introductory circuit theory by guillemin ernst

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ALCOMMUNICATION

CALENGINEERING

TEOFTECHNOLOGY

&SONS,INC.

ALL,LIMITED

P u b l i c D o m a i n , G o o g l e - d i g i t i z e d

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eofmu,tnot

withoutthe

ubli,her.

BER,1958

gCardNumber:53-11754

of America


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eenthusiastic

nspiration

P u b l i c D o m a i n ,

G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h

a t h i t r u s t . o r g / a c c e s s

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ader ofthisvolume,itshouldbe pointed

contemplatedsequence.Thesecondvolume

heapproximationpropertiesandusesof

nwithcircuitproblems,andwilllead logically

and Laplacetransformtheory,itscorrelation

ialequationviewpoint,anditsapplicationto

cedures.Theremainderofthisvolumewill

osynthesison asurveylevel,includingsome

andthecloselyrelatedtopicoftransmission

ctsof(linear,passive,bilateral)networkanal-

thesubjectofoneor twofinalvolumes.Work

beeninterruptedin favorofproceedingim-

cedpartwhichismoreurgentlyneeded.

title states,isintendedtobe anintroductory

ittheory—thetextforafirstcoursein circuits

tsmajoringinelectricalengineeringorfor

dagoodorientationalbackgroundinthesub-

astfive years'experienceingettingourE.E.

erightdirectionandour physicssophomores

ntationincircuitprinciplesand aflexible

feel thatcircuittheory(thatis,linear,

aleralcircuittheory—hereaftercalledjustplain

calengineer'sbreadandbutter,so tospeak.

ectwell beforehecantackleanyofthe other

andit isoftheutmostimportancethathis

mwith asetofbasicconceptsandwaysof

meobsoletethroughouttherest ofhisunder-

ars.Heshouldbestartedoff withthesame

ssesofanalysisthathewillbe usinginhis

sprofessionalworkfouror fiveyearslater.He

owell orbeableto usethemwiththesame

theshouldnever havetounlearnordiscard

ateron. Histhoughtsasasophomoreshould


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



sthat willfeedandsustainhis creativethinking

allife.Inotherwords,his firstcourseshould

oursebutthebeginningofa career.

erethe teachingofbasicconceptsandpro-

odistinctionshouldbemadebetweentheso-

he"advanced"methods.Werefertothings

solongaswe understandtheminsufficiently

makethemclearinsimpleterms.Oncewe

andclearly,itis nolongerdifficulttomakeit

inner.And,ifwedonot warnthebeginner

ableto distinguishwhenweareteachinghim

sandwhenthe"advanced."Suchadis-

theteacher'smind;tothestudentboth willbe

clear.

ngsbecausesometeachers,uponperusing

yconsidersomeof thetopicsdealtwith(as

thework)tobesomewhatmoreadvancedthan

propriateforsophomoreorjuniorstudents.

erinthisregardthata conceptisnotneces-

studentbecauseit happenstobeunfamiliar

llynoneofthematerialinthis bookisany

olvedinthe differentialorintegralcalculus

ppropriateforthesophomorelevel.Compared

ndofcircuitscourse,theworkis morechal-

salsofar moreinteresting.Tomystudents,

d reliablecritics,thereisnothingdrababout

eirenthusiasmandmoraleare high,andthe

itingtothem.This ishowthingsshouldbe.

ecificaboutthewaysinwhichthe intro-

ookdiffersfrommost.Primarilyit hits

remorefundamental,andattemptsinevery

sicideasand principlessoastopromoteflex-

emandfacileuseofthem intheirapplication

epracticalproblems.

outfundamentals,butsometimeswedon't

kethe matterofsettingupequilibrium

rcuit.Theveryfirststep istodecideupona

heymustbe independent,andmustbeade-

thenetworkatanymoment.Theusual

ofvariablesisto chooseasetofmeshcurrents

stoptoconsiderhowwecanbe surethat

andadequate,orwhethertheyare reversibly,


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uslyrelatedtothebranchcurrents?No.We

ndwealsotakeforgrantedthat thestudent

slysimple"matteroutforhimself.He

me,butrighthere hestoresupa lotoftrouble

ow untilmuchlaterinhis careerwhenhe

oxsituationandsuddenlydiscoversthathe

nfusingsituationexistswhenwe attemptto

variablesasinnodeanalysis.Thistopic,even

ergetsacross.Needlesstosay,I don'tthink

entalaboutthesethings.Ofcourse,our usual

snot averyimportantaspectofcircuittheory

dvancedtopicstoohighbrowforsophomores;

engineereverusesitanyway.Thislast re-

book.Ofcoursehedoesn'tuseit. Howcan

standwhatit'sall aboutandneverhadit

ownits possibilities?Asforthetopicbeing

res,thisisplain nonsense(towhichmysopho-

tlyattest).

entialofthisitemis concerned,letme

berofpertinentincidentsthatoccurredre-

rsconcernedwiththeBonnevillepower

cNorthwestwerehavingaconferencehere,and

wapproachto theanalysisproblemwhichis

uchpower-distributionnetworksandleadsto

onalprocedurethatbeatsusingtheoldnetwork

ew"approachconsistsinpickinganappro-

thelink currentswithloopcurrents,thetree

distributionsystemandthelinksbeingthe

urcesandloads.It seemsthatpoweren-

a morefundamentalapproachtocircuit

tialin gettingcloserandgivingmoreem-

heuse ofscalefactorsandtheprocessof

studentatthe outsetthatwearegoingto

earcircuits,but doweclearlyimpressupon

propertyor howwecancapitalizeonit?

ents,aswellas manyengineersinindustry,

cationsofthispropertyandofits usefulnessif

theconventionalprocedureinteaching

obscuresthisimportantaspectofthesubject

whatismistakenlyregardedasa"practical"


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ber,inmyown circuitscoursethatIattended

xcitationinthe numericalproblemswas

volts orsomeothervaluein currentpractice.

cher(andstillisby sometoday)thatwemust

fsuchpracticalvaluesofvoltage;thatit isan

onofanintroductorycircuitscoursetoen-

utthemagnitudesof significantquantities

sof todayarenotsostupid asallthat.

mmon"housecurrent"issuppliedat110and

nowthatthe frequencyis60cyclespersecond

nada),anda hostofotherpracticaldata

Furthermore,thesefactualdataabout

andarefar moreappropriatelypresentedina

ect.Itismuchmoreimportanttoemphasize

lt or1ampereasan excitationvalueisen-

eofany eventualityregardingsourcein-

this,weachievea certainsimplificationof

twehaveone lessfactortocarrythroughthe

nddivisions,andwebecomeeversomuch

mplicationofthelinear propertyofnetworks

e madebetweenpowercalculationsandvolt-

s,becausethenecessaryfactorsbywhichthe

emultipliedaredifferent.

eadvancedconcerningthespecificationof

eemstobeanurgentneed todootherwise,it

sume1 radianpersecondasthefrequency

earninghowhe cansubsequentlyadaptthe

yothervalue ofexcitationfrequency,thestu-

ppreciationofthe fundamentalwayinwhich

ponfrequencyasaparameter;andagaina

sgainedwithregardto thenumericalcom-

aloneis moreimportantasapractical

mightsuppose.Ihad occasionrecentlyto

reon aresearchprojectinanindustrial

tosuggestfrequencyscaling.Theensuing

withfantasticpowersof2*rand10, causingall

surdresults.Aprogramoffrequencyscaling

tparametervalues(criticalfrequenciesand

raightenedthings outinahurry.(The men

dentally,weretrainedasphysicists;sothe

durestoprovideasufficientlyclearunder-


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8apparentlynotrestrictedtoengineering

ueslieswithinthe frameworkofthesesame

ntimaterelationtofrequencyandamplitude

msinthis bookinvolveelementvalues

herange1-10.Here,again,criticswill argue

alisticandmaygive ourstudentsmistaken

acticalvalues.TothischallengeIreply:(a)

umb.(6) Theyhavelivedandwilllive in

heyhave ampleopportunitytofindoutwhat

concurrentlytakingacoordinatedlaboratory

thelpbutbecomeawareofthefactthat 1

acitance,(d)Itismuchmoreimportantfor

osesof calculation,wecansonormalizeour

ementvaluesintoarangewherepowersof 10

cedtoaminimum.Infact,it isthisnor-

swhataresometimescalled"universal

pertinentcircuitresponseunderawidevariety

consequencesoflinearitythatcannotbe

dditiveproperty(superposabilityofsolutions)

nandresponsefunctionsasa pairmaybedif-

nyfinitenumberoftimeswithouttheir

eotherbeingdestroyed.Butof utmostand

eproperdiscussionofandapproachto theim-

onnectionwecannotregardtransientanalysis

dealt withlateron.Transientanalysismust

a-csteady-stateresponseinorderthatthe

dancefunctionmayberecognized.Unless

onto circuittheoryisproperlyaccomplished,

hafalsenotionaboutthe impedancecon-

nlearnlateronbeforehecan acquireamentally

pedancereallyisand oftheomnipotentrole

To teachtheimpedanceconceptinitiallyin

formregardinga-csteady-stateresponse

understandingofitstrue natureandcausesa

effortthat wecannotaffordtoday.Inthis

bservedthatmanyofmy graduatestudents

steringtheimpedanceconceptthansomeofmy

ementalattitudeisnotpreconditionedbysome

edwhenwecouldregard thediscussionof


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itsasaluxury iteminourE.E.curriculum.

ctronic-controldevicesandtheincreasedim-

onlinksinourfast-movingmodernworldhave

leteasthe rotaryconverter.Adiscussionof

rcuitsisa mustinourpresentphysicsas well

east.Anditis wrongtothinkthatit logically

onfollowingtheintroductorysubject.With-

aturalbehaviorof atleastsomesimplecir-

esenttheimpedanceconceptbecausethe

equantitiesbywhichtheimpedanceisde-

importantconstantmultiplier.Theim-

atelyrelatedtothe transientbehaviorthanto

eresponse,althoughitcharacterizesboth.

eenthetransientandsteady-statebehavior

portantasa fundamentalprinciple,andwe

t fundamentalsunlessthisitemisdealtwith

isinterpretationof theimpedancefunction

requencyanditsgraphicalrepresentationin

ane.Throughthismeans,theevaluationofan

liedfrequencyisreducedtoa geometrical

ticalcasescanbesolvedby inspection,

bleapproximationsareallowable.Furtherex-

easleadsus, inalogicalmanner,tointerpret

theconstantsdeterminingthetransient

oallofthe practicallyusefulresultsordinarily

bleonlythroughuseof Laplacetransform

knowledgeaboutcircuitslieswithinthis

t,withoutquestion,itmaybe regardedasthe

y;yettheconventional"firstcourse"in

ted(withfewexceptions)makesnomention

itymaybe mentionedasanimportant

tshouldbeprominentthroughoutthedis-

roductorytreatmentofcircuittheory.Here

sedliterally,sincethe principleofdualityis

tivelybedisposedofbyaconcentrateddis-

eeminglyappropriatepoint,butinsteadisbest

nit againandagain,bringingouteachtime

taspectorapplicationofthisusefulconcept.

ructureofthisbook,it issignificanttopoint

ptersmayberegardedasa separateunit


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textforarathersolid subjectind-ccircuitsor

emedappropriate.Similarly,thesucceeding

a closelyknitunitthatcanbe usedseparately.

ookwaswritteninsucha waythatitcould

hetextfor aone-semestersubject,provided

ybeenexposedtoKirchhoff'slawsandsimple

physicscourse.Ifonlyone semestercanbe

ourphysicsstudents),then thismaterial

omise,whiletheavailabilityofthediscussions

ascollateralreadingmaterial(tobeconsulted

nylatertime) servesasastopgapin lieuof

allyadequatefoundationatthispointin the

erscanbedevotedtotheintroductorycir-

rs1through9 formanappropriatetext,and

offandgeneralizessomeofthe previousdis-

ateralreadingassignmentoras areminder

oryr eallyhasnoending.Inany event,the

with advancedworkinnetworkanalysisand

terialofChapter10 asanecessarybackground.

dualpurpose,asindeedithas servedduring

ment,theone-semesterversionbeingappropriate

dthe two-semesteronefortheE.E.'s.

tentialreader thatthisbookwillprove

oryasareferencework.Thusthediscussion

temlikeThevenin'stheorem,duality,the

cetransformations,etc.,willnotbefound

rtainpages.Discussionofsuchitemsaswellas

ndamentalprinciplesarescatteredthrough-

ntationhere,alittle morethere,andstillmore

kindofpiecemealpresentationisthatthe

edasatext,and thelearningprocessisa piece-

tanystageto havesomerepetitionofwhatwe

withtheadditionofafewnew ideas,followed

henbyfurther additions,etc.Another

sentationisthedualpurposethe bookismeant

essresultingfromtheseobjectivesIhopethe

herthan otherwise.

makesomespecificcommentsonthe

ptersandthe reasonsforitsparticularmode

wochaptersarethe resultofyearsofpractice

gabouthowbestto presentthesubjectof

quationsforanetwork,andwhy,inspite of


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sremainedsomuchconfusionandso littlecon-

ndaboutthis topic.AtlonglastI thinkI

his perplexingquestion,andChapters1and2

stheconventionalapproach(andIamas

followedit)attemptsto presenttoomuchat

nfusion.Thevariousmethodsusingtensoror

thesamedefect.Moreover,theyfailtodis-

mportantissueofnetworkgeometry,andin

ablefor anintroductorypresentation.

gequilibriumequationsinvolvesactually

allyrequirecarefulthoughtandconcen-

ding.Whenthesearesuperimposedtoform

tislittlewonderthat nothingbutmisunder-

kingresults.

ectinganappropriatesetof variablesand

betweentheseandthebranchvariables.It is

etworkgeometry(nomentionneednorshould

rchhoff'slaws,orthevolt-ampererelationsfor

es).Thetopicinvolvesa numberofsubtleties,

iresareasonablygoodappreciationofthe

sematterscanbe clarifiedeasilyifweex-

ngelseexceptthe purelygeometricalproper-

1.

,weareinaposition towriteequilibrium

ussionofthe Kirchhofflawsandhowtoapply

c.The thirdtopicconcernsthevoltrampere

andnowwecancombinetopics1, 2,3to

tionsintermsof thechosenvariables.Finally

urces,andourproblemofestablishingequi-

alprocedureofwritingKirchhofflaw

termsofloopcurrents.Herethefour steps

edinto thepotatonceand stirredtogether.

eofindigestion,unlesswesorestrictand

tureastorenderthe endresulttrivial.

lly,thatthediscussionsinChapter1 are

moresothanmightberegardedappropriate

torycourse.Inanswerto suchcommentI

wrotethechapter,Icouldseeno pointin

oreIhadfinishedwhatI hadtosayandwhat

mofnecessarymaterialtoformagood back-

ter. Topostponethediscussionofsomeof


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se,sinceasubsequentcontinuation(perhaps

havetorepeatparts oftheearlierarguments

nceinthepresentationas awhole.Idon't

fmoreinformationthanone carestoas-

ouldposeanyserious problem.Chapter1

rereadseveraltimes bythestudentat

tionalprogram.

ricalaspectsofduality,whichplayan im-

fChapter1,Ifound itconvenienttoinvent

o myknowledgeatleasthadnotpreviously

lofa cutsetIhave nameda"tieset,"and

"Thesenamesseemedmostappropriateto

willfindthemappropriatealso.

ftopics,allofwhicharedirectlyor indirectly

gtheprocessofobtainingsolutions.Syste-

ures,solutionbydeterminants,specialartifices

ypesofsymmetryprevail,shortmethods

res,wye-deltatransformations,sourcetrans-

atThevenin'sandNorton'stheorems

ytheorem(frequentlyaneffectiveaidinob-

knowledgeofhowpowercalculationsmustbe

ffectswhencausedbyseparatesourcesare

urrentsand voltageswhichare),thetrans-

errelationsinvariant,theequivalencerelations

dged-teeandlatticestructures—allthese

are dealingwiththebusinessofconstructing

elong togetherandthatitis usefultomake

while discussingtherestrictedcaseofre-

herearenoothercomplicationstointerfere

houghhere,asinChapter1, thetreatment

tmoreinclusivethanis essentialatanin-

sdifficultyneedtherebybecreated,sincethe

varioustopicscanalwaysbeappropriately

s madetohavethevarioustopicsintroduce

erthanbeforceduponthereader'sattentionin

,havingdiscussednetworkgeometry,and

ricalsetofequationsmaybe solvedbysys-

riables,whatismorelogicalthanforthe

aboutthegeometricalimplicationsofthis

eliminationofanodepotentialshouldcorre-

eeliminationofanode,and theeliminationof


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minationofthepertinentmesh.Suchacorrela-

ble,notonlyl eadslogicallytoapresentation

transformationsandtheirgeneralizations,but

ofdisagreeablealgebra,ascontrastedwithother

particularlyinthegeneral star-meshcase.A

fthereciprocitytheoremwhichlikewisefits

hesystematiceliminationprocedureis

gthatthesymmetryoftheparametermatrix

riumequationsisunchangedbyatypicalstep

esthevolt-ampererelationsfortheinduc-

ementsandshowsthatinductancenetworksand

ealtwithby meansofthesamemethods

etworks,isprimarilyconcernedwithadis-

dimpulsefunctions,intermsofwhichvarious

ctionsandswitchingoperationsmaycon-

connectionwiththeimpulsefunction,ithas

tiesinvolvedin itsinterpretationaretoo

ophomorelevelandthattheconceptis too

ercriticismisconsistentwithourprevailing

nvolvedinthedefinitionof theimpulseis

reasthat pertinenttotheformationofa de-

thecomprehensionofthissortoflimit

sophomore,thenweshallalsohavetogiveup

ferentialcalculus.

nreal,nothingcouldbefurtherfromthe

equentlyseethingsbumpingintoother

baseballforinstance.Theballchangesits

sin awink—andthat'sshortenough(com-

htoftheball)to benegligible.Forallpractical

tskineticenergyof flightinnotime. Ifwe

situationandsaythatthenonzerotime of

dandsowereally arenotdealingwithani m-

entobeconsistentweshouldbe equallyfussy

ecauseachangeinvalue(of aforceforin-

antlyeither;yetwenolongerobjectto step

ganalysis,becausewehavelivedwiththis

edtoit. Ourmathematicalmethodsofanalysis

zationofthetruestate ofaffairs,andtheim-

thingdifferentinthisrespectfromall theother

atweare accustomedtouse.

vingsingularityfunctionsofallordersis


/ h t t p : / / w w w . h




aythatwecannolongerneglectmakingour

matan earlystage.Mychiefreasonforin-

ellas thestepwhenIfirst wrotethistext

se Thevenin'sandNorton'stheoremswith

nceelementsinthetransient state.Sincethese

tiationandintegration,itwasclearthata

obedifferentiatedin thecourseofsolvinga

Todeprivethestudentof thisflexiblewayof

blems,Ifelt,wasnot inkeepingwithmybasic

thepresentationofsingularityfunctions

mintothesophomoreyear.

utthattheearlyintroductionof thesecon-

cuittheorydevelopsamoreopen-minded

studenttowardcharacteristicbehavior

ys,forexample,weweretold thatthe

sthadto becontinuous.Thoughthisistrue

s,itis muchbetternottomakesuchsweeping

structivetoshowthestudentthata dis-

producedinaninductanceonlythroughthe

pulsebutthatphysicalconditionsmaysome-

dofexcitationfunction.

nsient responseofsimplecircuits,making

ntionedabove.Theprimaryobjectiveistogive

erstandingoftransientresponsein first-and

therwithafacilewayofdealingwiththeperti-

onships,sothathewilldevelopaneasyand

oblemsofthis sort,ratherthanalwaysuse

ponderousandslowlymovingmachineryof

espectI haveseensomeawfulcrimes

ystudentswhohavelearnedtheLaplace

redeterminedtoLaplace-transformevery-

,andtheyget sotheycan'tsolvethesimplest

hinery.Theycan'twritedownthedischarge

sistancewithoutLaplace-transformingthe

twantanyofmystudentsto getintoafix

nowtheirsimpletransientsaswell asthey

dChapter5aimsto givethemthekindof

ishthisend.

apiecederesistanceasthe Frenchwouldsay.

soid,thenotionaboutcomplexfrequency,

tsinterpretationintermsofthenaturalfre-

phicalportrayalofthepole-zeropatternin


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mpedancesthroughgeometricalvisualization

interpretationofresonanceasanearcoin-

andnaturalfrequencies,reciprocalandcom-

magnitudeandfrequencyscaling,vectordia-

spectspertinenttothis generaltheme.Tran-

estirred togetherintoaprettyintimatemix-

unctionkeepingorderandclarifyingallof the

ps.Thecircuitsdealtwithareforthe most

oucheduponinChapter5so thatthestudent

wingthemathematicalstepswhilegetting

ceptsandmethodsof interpretationpresented

eelementcombinations,suchastheconstant-

ouble-tunedcircuits,arediscussedtowardthe

ertoshowthestudenthow simpleamatterit

onsintermsofthe ratherpowerfultoolswhich

eplacedathis command.

maldiscussionofenergyandpower relations.

alrestrictioninthederivationof pertinent

out-of-phasecomponentsofcurrentand

etodevelopa morephysicalappreciationof

hspecificattentiontothestoredenergyfunc-

einthe sinusoidalsteadystate,alongwiththe

onfunction.Thusthe definitionofreactive

e voltageandthequadraturecomponentof

withnophysicalpictureof whatthisquantity

dstobe considered.Whenitisseento be

ncebetweentheaveragevaluesofthestored

beginstobeappreciatedinphysicalterms.

dancesintermsofenergyfunctions,through

semeans,andthroughtheability thusto

quencycomputationthewholecourseoftheir

(forinstance,the determinationoftheim-

onancevicinityandcomputationofthefactor

glimpseofhowenergyandpowerconsiderations

broadersensethanmerelyforthecomputation

toprovidethemeansfordealingwith more

mcircuitsinthesinusoidalsteadystate than

sidered.Mostimportantinthisregardis the

ductivecoupling.Thetraditionalstumbling

mentofrandomsituations,namely,the

csigns,isovercomebyasystematicapproach


/ h t t p : / / w w w . h




itsuse forthecomputationofpertinent

ththeloopandnode bases.

ftransientresponseisgeneralized,first,

heso-calleda-ctransientsand,second,through

etesolutionforanyfinitelumpednetwork,

enticalinform with,butmuchmoresimply

elyobtainablethroughLaplacetransform

ycomplexintegration.Itisinthesediscussions

exfrequencyisfullydevelopedandillustrated

xactcoincidencebetweenexcitationand

ectresonance).Itisshownhowallthemany

yderivedonlybyFourierandLaplacetrans-

ndrigorouslyestablishedbyinspectionofthe

egeneralcase,andthesetheorems(orproper-

)arediscussedand illustratedbymeansof

structedbystartingfromassumedpole-zero

redtransferfunctionsandsynthesizingthe

forthefirsttimein thehistoryoftextbooks

eaderis presentedwithillustrativeexamples

nd-ordersystems.Hewillfindamultiple-

hanthe hackneyedRLCcircuitforthecriti-

ewillfindexamplesthat arerepresentativeof

ristics,aswellasillustrativeofthetheoretical

m.

esisitwas notpossibletoconstructreally

mples.Ifacircuitwithmorethantwo or

ed,thesolutionofa characteristicequationof

telyinvolved,andtheresultingrandomchar-

nedaftermuchdisagreeableworkwashardly

ginteresting.Beingabletostartfroma pole-

th directions(toanetworkonthe onehand,

nseontheother)opensupa hostofpossibilities

he textbookwriterofthepast.Withina

ethemostof thissituationinworkingouta set

rChapter9.

previously,suppliesacertaingeneralityand

ationofequilibriumequationsandenergy

scussedalreadybuthavenot beenestablished

readerreachesthischapter,hewillbe

containsexceptthemathematicalmethods

rfectlygeneraland yetcompactandconcise


/ h t t p : / / w w w . h




ticaldiscussionoftheprincipleofduality

efrom itinthel ightofthebroaderviewpoint

fnetworktheoryis,of course,nowherenearly

t,sinceonevolumecannotcontainall ofit,

ablepointatwhichtostop.

ethingabouthistoricalnotes(whodid what,

encestosourcematerialandall that,because

n'tdoneanyof thissortofthing.As amatter

orksofKirchhoff,Helmholtz,Cauchy,Lord

wothersofsimilarstandingand vintage,there

ededtoestablishthebackgroundfornetwork

einclinationto"dosomedigging"(mostof

etouthistoricalfacts,hewillhaveno diffi-

phicalhelpandtheencouragementfromhis

o notmeantobelittlethe importanceof

ckgroundontheevolutionofscienceand

rktheory),butthewherewithaltogointothis

available.I wouldratherconfinemylimited

wstheyarelimited!)tomakingavailablethe

vailable.

chingofthissubjectIregard itasim-

entfrequentlythatnetworktheoryhas a

tionwiththeprincipleofduality);itis a

hing;it istwo-faced,ifyouplease.There

ect:the physicalandthetheoretical.The

sentedbyMr.Hyde—asmoothcharacterwho

andcan'tbetrusted.The mathematical

yDr.Jekyll—adependable,extremelyprecise

pondsaccordingtoestablishedcustom.

eorythatweworkwithonpaper,involving

ytheones specificallyincluded.Mr.Hyde

eetin thelaboratoryorinthe field.Heis

mentsunderhisjacketandpullingthem out

gtime.Wecan learnallaboutDr.Jekyll's

ableperiod,butMr.Hydewillcontinuetofool

nd oftime.Inorderto beabletotacklehim

mewellacquaintedwithDr.Jekyllandhis

lmostwhollyconcernedwiththelatter.I

deto theboysinthe laboratory.

sinthe laboratory,"thatistosay,the

whoassistinadministeringthis materialto

ts,Iwishhereto thankthemoneandallfor


/ h t t p : / / w w w . h




tionsandtheirenthusiasticcooperation

is"five-yearplan."I cannotnameonewith-

cannotnamethemallbecauseIcan'tbe sure

o.Sothey'llall havetoremainnameless;

gonly.It won'tbelongbeforeeachonemakes

mehavealready.

ishyoualla pleasantvoyage—throughthe

reveryoumay begoing.


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/ h t t p : / / w w w . h




NetworkVariables5

NetworksS

k5

7

rrents;TieSetsand Tie-SetSchedules./10

irVoltages;CutSetsandCut-SetSchedules17

ChoosingCurrentVariables23

ChoosingVoltageVariables33

1

s64

heKirchhoffLawEquations68

nsontheLoopand NodeBases71

heLoopandNodeBases77

ryofParameterMatrices79

hatAreAdequateinManyPracticalCases81

uresforDerivingEquilibriumEquations96

RelatedTopics112

tionMethods112

6

adderandOtherSpecialNetworkCon-

ns;Wye-Delta(F-A)Equivalents127

sTheorems138

m148

ferFunctions153—

igurationsandTheirEquivalenceRelations161

ansformationsunderWhichTheyRemain


/ h t t p : / / w w w . h




rceFunctions188

elationsoftheElements188

rces190

yFunctions;SomePhysicalInterpreta-

ations203

rParametersofSimpleGeometricalConfigura-

onResponseofSimpleCircuits222

t;GeneralPropertiesof theSolution222

thematicalandPhysicalAspects230

s;TheVenin'sandNorton'sTheoremsand

LCircuit241

43

LCCircuit251

ryInitialConditions253

ransientResponseofOne-,Two-,andThree-Ele-

tsinthe SinusoidalSteadyState270

Sucha PredominantPartintheStudyofElec-

onofSinusoids273

mpedanceConcept282

anceintheComplexFrequencyPlane286

anceFunctionsforSimpleCircuits289

sonance297

arFormsofImpedanceandAdmittance

nterpretationofResonance301

mentaryImpedancesandAdmittances305

cyScaling309

anceFunctions;TheirPropertiesandUses315

eSinusoidalSteadyState340

geElements340

ementsWhen VoltageandCurrentAre

onsina CompleteCircuit343

wer;VectorPower348

EffectiveValues352

ceinTermsofEnergyFunctions354

rgyFunctionsforMoreComplexNetworks357

ples358


/ h t t p : / / w w w . h




nthe SinusoidalSteadyState366

quilibriumEquations366

es371

HowtoDealwith It374

80

EquationsWhenMutualInductances

-PointandTransferImpedancesforLad-

ymmetryinStructureandSourceDistri-

ts388

gwithSteady-StateandTransientBe-

rcuits401

ewithAlternatingExcitation401

heConceptsofComplexFrequencyand

mains414

orAnyFiniteLumped-ConstantNetwork419

riumEquationsforDriving-Pointand

rocityAgain426

lSolution431

0

ferFunctions462

ns468

tEquationsandEnergyRelations483

a483

ricesandVolt-AmpereRelations491

ntheNodeBasis496

ntheLoopBasis499

502

fandLagrangeEquations520

unctions522


/ h t t p : / / w w w . h




/ h t t p : / / w w w . h




nthisbook,andthose intheonesto follow

mplestclassofelectricalnetworks,thereader

will findthemtobeeither simpleorrestricted

mportance.Regardingtheirimportance,it

understandingofthetheoryofthissimplest

spensableprerequisiteto thestudyofall

ssignificanttoobservethatbecauseoftheir

evelopthetheoryofthisclassof networksto

mpleteness.Asaresult,thistheoryplays a

and developmentofalmostallelectrical

sthereforeasfundamentaltothe intellectual

lengineerasisa knowledgeofmathematics

marymissionofthis introductionisaccom-

agraphsareintendedtoprovidetheunin-

anideaas towhatanelectricalnetworkis,

lassofnetworksmentionedabove.Actually

tthereaderwhois totallyunacquaintedwith

hbenefitfroman exposuretosuchadefinition

rstandthemclearlyonly afterhehasgained

dinnetworktheory.Ontheotherhand,such

eaderwithasufficientinitial orientationto

erperspectiveasheprogresseswiththestudies

aracteristicsofalargeproportionofall

quatelydescribedthroughaknowledgeofcur-

functionsat appropriatelyselectedpointsor

behaviorofanelectronicamplifier,for

intermsofitsvolt-ampererelationsatspecified

pairs;the performancecharacteristicsofa

stributionof electricenergyorforthecon-

srepresentingcodedinformationareexpressible

and currentvaluesatappropriatepoints


/ h t t p : / / w w w . h




rof amotor-generatorsetisconveniently

tageandcurrentinputtothemotor relative

outputfrom thegenerator;theelectrical

arylightbulbareadequatelydescribedin

ntrelationsatitsterminals.

therfeaturesbesidesthe electricalonesare

re, forexample,themechanicalphenomena

fthe motor-generatorset,orthelightspec-

tbulb referredtoabove.Aseparationofthe

elyelectricalstudiesinsuchcasesis,however,

alwaysbeaccomplishedunderanappropri-

mentalconditions.Itmayadditionallybe

yingapproximationsandidealizationsinorder

turesofthe problemmanageableinreasonable

e,the resultingrepresentationoftheoriginal

ibedbytheterm"electriccircuit"or"network."

aythusbean idealizedorskeletonized

tricallyrelevantfeaturesofsomephysicalunit

acteristicsareonlyincidentalorat most

rollingitsstructureandbehavior,thereare

ethecircuitisthewholedeviceandits function

tinalarger system.Theelectric"wave

tworks"essentialtolong-distancetelephone

the"controlnetworks"inservomechanisms

ere theelectriccircuitnolongerplaysan

placealong withotherimportantelectro-

devicesasahighlysignificantunitorbuilding

essfuloperationofmodernpower,communica-

onthevolt-amperebehaviorofan electric

geandenergy-dissipationproperties.Energy

electricandmagneticfieldsassociatedwiththe

sipationispracticallyever-presentbecauseof

owofelectricchargethroughconductors.

ominateinmoldingtheelectricalbehaviorofa

associatedfieldsandthedissipativecharacter

aths.Althoughtheireffectsarephysically

tanyactualdevice,theidealizationreferred

tsonetoassignthem toseparateportionsof

regardthese portionsashavingnegligible

aksofcertain"lumped"partsashaving

one,othersashavinginfluenceonlyuponthe


/ h t t p : / / w w w . h




s,andathirdgrouprelatedsolelyto theper-

s thelumpedparametersorelementsofa

nds:the resistanceparameterordissipative

parameterwhichisrelatedtotheassociated

apacitanceparameterappropriatetothe

hysicalembodimentsofthesenetworkparam-

ngwherevertheiroccurrenceisdeliberate

familiartothe readerasresistors(usually

ngpoorconductivity),inductorssuchaswire'

uentlyintheformofmetallicsheetsor plates

finsulatingmaterial).Itisimportantto

alembodimentsarenotexactrepresentations

mentswhich,bydefinition,are"pure"in the

nsnoneofthe othertwo.Inanyphysical

einductiveandcapacitiveeffectsareunavoid-

apacitiveeffectsina physicalinductor,etc.

edeffectspresentinphysicalresistors,induc-

mmonlyreferredtoas"parasitics."Since

sknownparasiticelementscanalwaysbe

degreeofapproximationintermsof theo-

methodofcircuitanalysisbaseduponpure

quateanduseful.

eacrossanelementtothe currentthrough

rredtoasits pertinentvolt-ampererelationship,

ne(throughoutreasonableoperatingranges),

antofproportionalityisdesignatedasthe

thevaluesof networkelementsarefunc-

themor ofthecurrentcarriedbythem.For

oilrepresentsaninductanceelementwhose

ecoilcurrent;an electrontuberepresentsa

ththeappliedvoltage.Suchelementsare

sethevoltageis notlinearlyproportionalto

tothecurrent derivativeorintegral,which-

portanttodistinguishnetworksthatcontain

thatdonot,and torecognizesignificantdiffer-

aracteristics,forthesedifferencesformthe

tionofspecifictypesofelementsis madein

s.

nearaswellas nonlinear,whosevoltageor

ertiesdependupontheirorientationwith


/ h t t p : / / w w w . h




xcitationandobservation.Thesearespoken

esor elements;andwherevertheusualones

omthese,theyare referredtoasbilateral

tionhavingabearinguponnetworkbe-

owhetherthenetworkdoesor doesnotcon-

straintsotherthanthoseexplicitlygivenby

Ifitdoes,thenonemay expectattimesto

ne putsintothenetwork,orto obtainacon-

eabsenceofa powerinput.Whenanetwork

rgysourcesand/orconstraints,itiscalled

redtoas beingpassive.

passivebilateralnetworkisthesimplest

ysisneededinastudy ofitsbehaviorunder

ons.Toanintroductoryunderstandingofthe

alaspectsofthistypeofnetwork,thediscus-

eare directed.


/ h t t p : / / w w w . h




works

redistinguishedfromoneanotheraccording

hatareinvolved,andinthemannerof their

vennetworkconsistingofresistanceelements

sistancenetwork;andinductanceorcapacitance

nedassuchinwhichonly inductancesor

d.Nextinorderofcomplexityaretheso-called

preciselytheLCnetworks(thosecontaining

nceelementsbut,byassumption,noresistances),

nductiveeffectsareabsent,andRLnetworks

sareabsent.TheRLCnetworkthenrepre-

e categoryoflinearpassivenetworks.

of elementsinvolvedinagivennetworkis

ofnetworkgeometrythatconcernsitselfsolely

he variouselementsaregroupedandinter-

als.Inorderto enhancethisaspectofanet-

onefrequentlydrawsaschematicrepresentation

nisasyet madebetweenkindsofelements.

esentedmerelybya linewithsmallcirclesat

ls.Suchagraphicalportrayalshowingthe

onofelementsonly,iscalleda graphofthe

owsanexampleofa networkasitisusually

thevariouskindsofelements[part(a)] and

pearswhenonlyitsgeometricalaspectsare

(b)]. Thenumbersassociatedwiththe

edfortheiridentificationonly.Theterminals

commontotwoor morebrancheswhere

ferredtoas nodes.


/ h t t p : / / w w w . h




DNETWORKVARIABLES

ment

element

candits graph.

chvariousparts ofanetworkareonly

npart(a) ofFig.2wheretwopairs ofmutually

volved.Herethecorrespondinggraph(shown

tsofthree separateparts;anditis seenalso

orkgraph

raphof anetworkconsistingofseveralseparateparts.

ytheterminusofasingle branchaswellas the

ralbranches.

rktherearethusassociatedthreethingsor

es,nodes,andseparateparts.Thegraphisthe

tainsonly

suseful

houldbest

enetwork

esand

ethera

blesare not

adequate

coa-forthe uniquecharacterizationofthe

f anetworkatanymoment.

tthatan

nsituationslike theoneinFig.2 through


/ h t t p : / / w w w . h




E"

hofthe separatepartstobecomecoinci-

rts,as isshowninthe graphofFig.3.

superimposednodesareconstrainedto

ential,norestrictionsareimposeduponany

urrentsthroughthis modificationwhich

ofnodesand thenumberofseparatepartsby

bsequentdiscussionsitis thuspossible

oconsideronly graphshavingoneseparate

cesin evidenceanumberofclosedpaths

rculate.Thispropertyofagraph(that it

viouslynecessarytotheexistenceofcurrents

ssibletrees(solidlines).

Itis apropertythatcanbedestroyedthrough

chosenbranches.

ennetworkis showninpart(a),andagain

meofthe branchesrepresentedbydotted

swereremoved,therewouldremainin

(b) andin(c)a graphhavingallofthe nodes

utnoclosedpaths.Thisremnantof the

tree"forthereasonthatits structure(like

the significantpropertyofhavingnoclosed

definedasany setofbranchesinthe original

tinnumbertoconnectallofthe nodes.Itis

snumberisalwaysnt— 1wherentdenotes

For,if westartwithonlythenodes drawn

rthatthefirstaddedbranchconnectstwo

additionalbranchisneededfor eachnodecon-

minimumnumberofnt— 1branchesare


/ h t t p : / / w w w . h




DNETWORKVARIABLES

nodes,thenitis likewiseclearthattheresulting

edpaths,forthecreationofa closedpath

nodesthat arealreadycontacted,andhence

anchesthanare actuallyneededmerelyto

t ispossibletodrawnumeroustrees,since

snot auniqueone.Eachtree,however,con-

ndconsistsof

redto,inanygivenchoice,asthe treebranches.

ketheonesshowndottedin parts(b)and

ks.IfthereareI ofthese,andifthe totalnum-

workgraphisdenotedbyb,then evidently

lrelationtowhichweshallreturnin thefollow-

ofa networkiscompletelyknownifthe

nall ofitsbranchesareknown.Thebranch

atedtothebranchvoltagesthroughfunda-

racterizethevolt-amperebehaviorofthe

stance,inaresistancebranchthevoltage

sthecurrentin thatbranchtimestheper-

nacapacitancebranchthevoltageequalsthe

uetimesthetimeintegral ofthebranchcur-

ebranchthevoltageisgivenbythetime

ththeinductanceasa proportionalityfactor.

edrelationsbecomesomewhatmoreelaborate

sinthenetworkaremutuallycoupled(aswill

,their determinationinnowayinvolvesthe

onoftheelements.Onecanalways,ina

elatethebranchvoltagesdirectlyandre-

ents.

deitherthebranchcurrentsaloneor the

dequatelycharacterizingthenetworkbe-

of branchesisdenotedbyb,thenfrom either

uantitiesthatplaytheroleof unknownsor

finding thenetworkresponse.Weshallnow

uantitiesisnot anindependentone,butthat


/ h t t p : / / w w w . h




haracterizethenetworkequilibrium,whether

ebasis.

eisselected,thenthe totalityofbbranches

ps:the treebranchesandthelinks.Corre-

rrentsareseparatedintotree-branchcurrents

emovalor openingofthelinksdestroysall

forcerendersall branchcurrentszero,it

tofsettingonly thelinkcurrentsequaltozero

tworkto bezero.* Thelinkcurrentsalone

death,sotospeak,overthe entirenetwork.

rentvalues;thatis,itmust hepossihleto

hraT'f'ncurrantsuniquelyintermsof thelink

fromthis argumentisthat,oftheb branch

I areindependent;Iisthe smallestnumberof

all otherscanbeexpresseduniquely.This

lowfrom thefactthatallcurrentsbecome

sare zero.Thusitis clearthatthenumberof

relynotlargerthan I,for,ifoneof thetree-

medalsotobeindependent,thenits value

zerowhenallthe linkcurrentsaresetequal

s manifestlyimpossiblephysically.Itis

and thatthenumberofindependentcurrents

then itwouldhavetobe possibletorender

kzerowithoneormorelinks stillinplace,and

becauseclosedpathsexistsolongas someof

t mustbepossibletoexpressuniquelythe

sofIvariablesalone.Aswill beshownlater,

yappropriatesetof linkcurrents(according

eforatree), butmoregenerallytheymaybe

wayssothatnumerousspecificrequirements

ardthebranchvoltagesasseparatedintotwo

oltages,andthelinkvoltages.Sincethetree

enodes,itis clearthat,ifthetree-branchvolt-

s notnecessarythatweconcernourselveswiththe

orkisenergizedalthoughsomesort ofexcitationisimplied

geswouldotherwisebezero,regardlessof whetherthe

thereaderinsistsuponbeingspecificaboutthe nature

ctureinhis mindasmallboytossing coulombsintothe

ntervals.


/ h t t p : / / w w w . h




ANDNETWORKVARIABLES

(throughshort-circuitingthetreebranches,for

epotentialsbecomecoincident,andhenceall

dtobezero.Thus,the actofsettingonlythe

altozeroforcesallvoltagesin thenetwork

hvoltagesaloneholdthe poweroflifeand

eentire network.Itmustbepossible,there-

nkvoltagesuniquelyintermsof thetree-

tagesina networkareindependent,

othebranchesofa selectedtree.Surelyno

nbe independentbecauseoneormoreofthe

avetobe independent,andthisassumption

thatall voltagesbecomezerothroughshort-

esalone.Ontheotherhand,no smallernum-

m thecontrollingset,foritis physicallynot

nodepotentialstocoincidesolongas some

ainnonzero.

at thestateofanetworkcanuniquelybe char-

sof I=6— nt+1c urrentsorbyn=nt— 1

rentsmay,forexample,beanyset oflink

maybeanyset oftree-branchvoltages.

echaracterizationofanetworkintermsof

adifferentnumberofunknownsthandoesits

ofvoltagevariables.Thereisnothingincon-

onsinceweareat presentconsideringthe

eamongvoltagesorcurrentsfromageometrical

cally,thenumberofindependentvariables

ysicalsystemdeterminesuniquelyitsso-called

numberdependsneitheruponanyalgebraic

ponthemannerin whichthevariablesare

appropriatetoraise thesequestionsatthis

momentconsideringonlythosefeaturesofour

dbythegeometricalaspectsofthe given

rents;TieSets andTie-Set

inkcurrentsan interestinggeometrical

lwhentheseare selectedasasetof variables.

resentedin termsofaspecificexample.In

enetworkgraphandinparts(b), (c),and(d)

veralpossiblechoicesforatree.Forthe tree


/ h t t p : / / w w w . h




URRENTS

umbered1,2,3,4are thelinks.Ifoneof

ree,theresultingstructurehasjustone closed

erentforeachlink.Thus,forthischoiceof

edpathsisassociatedwiththerespectivelinks.

atedbylooparrows,numberedtocorrespond

nkswithwhichthey areassociatedand

possibletrees.

heselooparrows)so astobeconfluentwith

s.Thusloop1is formedbyplacinglink1

5(b);loop2is formedbyplacinglink2alone

hatwemaygiveto thelinkcurrentsanew

atofbeingcirculatorycurrentsorloopcurrents.

s corresponding,respectively,tothethreetrees shown

entifiedwithaloopcurrent;theremaining

learlyexpressibleasappropriatesuperpositions

henceareuniquelydeterminedby thelink

ier.

henetworkgraphofFig.5(a)are denoted

dtocorrespondtothebranchnumbering,and

graphofFig.6(a) aredenotedbyii, i2,it,i4t

tifications

=U(3)


/ h t t p : / / w w w . h




ANDNETWORKVARIABLES

gs.5(a)and6(a)onecan thenreadilyexpress

hcurrentsasappropriatesuperpositionsof

tions 3,have

xpressthetree-branchcurrents,uniquely

msofthelink currents.Thus,oftheeight

phof Fig.5(a),onlyfouraregeometrically

reappropriatetothesetof linksassociated

thetree ofFig.5(b),thelink currentsare

5(c)tlieyareji,h,j5,h-Herewemay

U(6)

ateonthecontoursindicatedinFig.6(b),

ughinserting,oneat atime,thebranches

ig.5(c).Thetree-branchcurrentsinthis

softhe loopcurrentsbytherelations

tionofFigs.5(a) and6(b)throughnotingthat

anchesresultfromthesuperpositionofper-


/ h t t p : / / w w w . h




URRENTS

s.6into 7,oneagainobtainsthetree-

dintermsofthelink currents

morethefactthatonly fouroftheeight

etricallyindependent.

ainstconcludingthatanyfourof theeight

gardedasanindependentset.The branches

pendentcurrentsmustbethelinksassociated

umstancethatassurestheindependenceof

chcurrentsjs,ja, jV,j$,forexample,could,

ntcurrentsbecausetheremainingbranches

ee.Theconceptofatreeis recognizedasuseful

nd unambiguousmethodofdecidingwhether

currentsis anindependentone.Oronecan

rovidesa straightforwardmethodofdeter-

dependentcurrentvariablesforanygiven

lanotherpossiblechoicefora treeappropri-

,andin Fig.6(c)isshownthecorresponding

nehas

ii (9)

therefollowsthat

— Js

— ji— js

kshavinglargenumbersofbranchesand

egeometries,onemusthavea lesscumbersome

edureforobtainingthealgebraicrelationships


/ h t t p : / / w w w . h




DNETWORKVARIABLES

ntsandtheloop-currentvariables.Thusitis

eprocessofdrawingandnumberingthe

ops,andsubsequentlyobtainingbyinspection

nsforthebranchcurrentsasalgebraicsums

canbecomebothtediousandconfusingin

exgeometries.

tingtheloopsassociatedwiththeselection

hroughuseofa~schedulesuchas11,which

g.5(a)withthetree ofpart(c)andhence

6(b).Tointerpretthisschedulewenote that

oop 1,indicatesthatacircuitaroundthis

singin thepositivereferencedirection,

n thenegativereferencedirection,branch6.

nchesparticipateinformingthecontourof

spondingspacesinthefirstrowofthe schedule

esecondrowissimilarlyconstructed,noting

touris formedthroughtraversingbranches

branch8negatively.Thusthesuccessive

atetheconfluentsets ofbranchesthatpartici-


/ h t t p : / / w w w . h




URRENTS

ds theequations

—

th Eqs.6and7.

ulehasthe propertyjustmentionedmay

osingthati tisoriginallyconstructed,by

relationshipsexpressedinEqs.12.Onesubse-

hytheresultingrowsofthe scheduleindicate

,throughnotingthatthe nonzeroelements

thbranchestraversedbythesameloopcur-

ymustformtheclosedpathin question.

theschedulemaythusbedone ineitherof

cordingtoaset ofindependentclosedpaths

atedwithaselectedtree),orbycolumns,

onsexpressingbranchcurrentsin termsof

edbycolumns,therowsofthe schedule

eclosedpathsuponwhichtheassociatedloop

onstructedbyrowsfromagivenset ofclosed

esultingscheduleautomaticallyyieldthe

branchcurrentsin termsoftheloopcurrents.

chforreasonsgivenlater iscalledatie-set

tobea compactandeffectivemeansforindi-

alstructureoftheclosedpathsand theresult-

weenbranchcurrentsandloopcurrents.

p,onemayinitiallybeconcernedaboutits

efewerloopcurrentsthanbranchcurrents.

12fortheloopcurrentsi ntermsofbranch

zledbythe factthattherearemoreequations

,thenumberofindependentequationsamong

mberofunknownloopcurrents(forreasons

ussion),andtheequationscollectivelyform

ethedesiredsolutionis effectedthroughsep-

dependentsubsetandsolvingthese.Knowing

ginallyobtainedthroughchoiceofthetree

ngbranchcurrentsj\,j2, js,j\asa possible

sthatthecorrespondingequationsamongthose

dedasanindependentsubset.Theseyieldthe


/ h t t p : / / w w w . h




ANDNETWORKVARIABLES

v,i3= js,{4=jV asindicatedinEqs.6

lthatthe independentsubsetchosenfrom

one.Thus,ifweconsiderthetree ofFig.5(d)

omesclearthatbranchcurrentsj4t js,j7,js

ecorrespondingequationsseparatedfrom12,

rdedasanappropriateindependentsubset.

+j5=ji, andthatj4— j7~js=J2, it

htheformerresult.

reindependent.Asimplerulefor picking

ochoosethosecorrespondingtothe linkcur-

ssibletree.Anyfour independentonesmay

pcurrents.Substitutionofthesesolutions

nsthenyieldsthe previouslydiscussedrela-

hcurrentsandlinkcurrents.

tyin understandingthissituationsincethe

adeitamplyclearthatthe linkcurrentsorloop

ntsetandall otherbranchcurrentsareuniquely

sETareconsistentwiththisviewpointand

d explicitrelationspertinentthereto.Hence

be unique,nomatterwhatspecificapproach

end.

1maybe constructedeitherbycolumnsor

ntwillbe thatitis constructedbyrowsfrom

tsof confluentbranchesformingthepertinent

eplacedin evidence,onebyone,through

areopenedexceptone,thusforcingallbut

entstobezero.The existenceofasingleloop


/ h t t p : / / w w w . h




PAIRVOLTAGES17

ranchesformingthe closedpathonwhich

s.Thissetof branches,calledatieset,is

nthepertinent rowofthetie-setschedule!

workgraphpermitsitsmappabilityupona

withoutcrossedbranches,thenwemayregard

undarythatdividesthetotal networkintotwo

nchesinsucha setareimaginedtoshrink

ducetoasingle point,thenetworkbecomes

fishnetwouldby meansofadrawstring),

dedbythe tiesetbecomeeffectivelyseparated

e.Itis thisinterpretationofthetie setthat

importantvariationsinthisprocedurefor

esetofcurrentvariables,weshall leavethese

andturnourattentionnowto thealternate

ust described)offormulatingasetofnet-

ebasis.

rVoltages;CutSetsandCut-Set

networkpicture,anentirelyanalogous

ebeginbyregardingthe tree-branchvoltages

endentvariablesintermsofwhichthestate ofa

expressed.Sincethetreebranchesconnect

bletotracea pathfromanynodetoany other

versingtree branchesalone;andthereforeit

differenceinpotentialbetweenanypairof

-branchvoltagesalone.Moreover,^thepath

sviatreebranchesisuniquesincethe treehas

offersnoalternatepaths betweennodepairs.

fferencebetweenanytwonodes,referredto

voltage,isuniquelyexpressiblein termsofthe

elinkvoltages,whichareaparticularset

thusrecognizedtobeuniquelyexpressiblein

oltages.

ipleswiththe networkgraphofFig.5(a),

egivenin part(b)ofthis samefigure.Ifthe

edbyvitv2,. ••,t>s,numberedtocorrespond

ering,thenthequantitiest>5, v6,v7,v&arethe

eon asphere(forexampleonethatrequires adoughnut-

trotalltie setshavethisproperty.Thispointis discussed


/ h t t p : / / w w w . h




ANDNETWORKVARIABLES

hencemayberegardedasanindependentset.

beregardedasnode-pairvoltages,and,since

osensetofvariables,wedistinguishthem

tationandwrite

7, e4=t>s(15)

eparallelstheuseofaseparatenotationfor the

whenchoosingvariablesona currentbasis.

dentifiedwith loopcurrents;inEqs.15the

dentifiedwithnode-pairvoltages.

ages,namelythelinkvoltages,arenow

softhe fourtree-branchornode-pairvoltages

ig.5(a)we have

e2

e3

4

i

eseequationsis toregardeachlinkvoltage

etweenthenodesterminatingthepertinent

ofthesenodestothe otherviatreebranches

theseveraltree-branchvoltagesencountered.

osen,thebranchvoltagesv3,v4t va,vS

dependentset,andwemaketheidentifications

6e4=t>s(17)

kvoltagesin termsoftheseread

+e3— e4

— ei—63+64

s.16 and18bearout thetruthofa state-

effectthatanyset oftree-branchvoltages

ependentgroupof variablesintermsofwhich

ages(linkvoltages)areuniquelyexpressible.

5,anytreehasfour branches.Hence,of

onlyfouraregeometricallyindependent.


/ h t t p : / / w w w . h




PAIRVOLTAGES

tinentto anyselectedtree;andtherestare

ofthem.

exnetworkgeometriesitbecomesusefulto

cedurefortheselectionofnode-pairvoltage

eexpressionofthe branchvoltagesinterms

mentofthisendfollowsapatternthat is

al)tothatdescribedin thepreviousarticle

s tosay,weseektoconstructa schedule

basisin thesamewaythatthetie-setschedule

basis.Tothisendwe mustfirstestablishthe

nforasetofbrancheswhich,forthevoltage

usto thatdefinedforthecurrentbasisby atie

chesforminga closedloop).Thelatteris

hopeningallofthe linksbutone,sothat all

eptone.Theanalogousprocedureona voltage

eofthe node-pair(i.e.,tree-branch)voltages

mplishedthroughshort-circuitingallbutone

ctwill ingeneralsimultaneouslyshort-

t therewillinanynontrivialcasebe left

heonenonshort-circuitedtreebranchthat

uitedandwillappeartoform connectinglinks

erminatingthepertinenttree branch.This

alleda cutset,isthedesired analogueofatie

edelaborationwillclarify.

kofFig.5(a)and thetreeofpart(b) of

epertinentstipulationof node-pairvoltages

hecut-setscheduleappropriate'to thissitu-


/ h t t p : / / w w w . h




ANDNETWORKVARIABLES

henode-pairvoltagesei,ej.,e3,e4as rises,

aredrops.Forthis reasonthereferencearrow

forthe vthatitis numericallyequalto.With

siderindetailthe constructionofschedule19

hispurposetoFig.5.Since byEqs.15,ei= u5,

alsofbranch5 constitutenodepair1,andthe

rrowforeiisat thetailendof thereference

ousremarksapplytothe otherthreenode

alsof branches6,7,and8.

sociatedwithnodepair 1,weregardei

pairvoltage;thatis, branches6,7,and8are

seconditionsitshouldbeclearthatlinks2

cuited,butthatlinks 1and4togetherwith

hort-circuited.Thesethreebranches,there-

pertinentto nodepair1,andthe corre-

rstrow ofschedule19thusarethe onlynon-

signsofthesenonzeroelements,wenotethata

rrentsthat areconfluentwiththereference

d5,andcounterfluentwiththereferencearrowin

tothenonzeroelementis chosenpositivefor

veforcounterfluence.

entifiedwiththeterminalsofbranch6.

sociatedcutset,weimaginethe othertree

ited,whencethenonshort-circuitedbranches

onventionjustdescribedyieldsplussigns for

minussignforbranch2.Constructionofthe

setschedulefollowsthesamepattern.

mnsinthisscheduleascontainingtheco-

tensystemofequationsexpressingthe branch

ode-pairvoltages,wehave

tsexpressedbyEqs. 15and16.Hencewesee

9couldalternativelyhavebeenconstructed

qs.15and16, ortheequivalentEqs.20.

fbranchesthatbecomesenergizedwhenits

eistheonlynonzeroone amongallofthenode-


/ h t t p : / / w w w . h




PAIRVOLTAGES

esetisthat groupofbranchesthatbecomes

ntloopcurrentis theonlynonzerooneamong

sclear,therefore,thatthe cutsetsmay

mEqs.20throughpickingout thebranches

nzerov'sthatresultfrom consideringthee's

e;andit isthusappreciatedthattheconstruc-

maybe doneeitherbyrowsaccordingtothe

ycolumnsaccordingtoa setofequations

onstructedbyrows,throughpickingoutthe

escheduleautomaticallyyieldtheassociated

oltagedropsintermsof thenode-pairvoltages;

mns,therowsofthe scheduleautomatically

ts.

calcharacteristicofacutset isrecognizedfrom

n.(Ifallbutoneof thebranchesinatreebe-

nlythetwonodesat theendsofthenonshort-

Orwemaysaythatthetotality ofnodesin

twogroups;allthe nodesofonegroupbecome

ofthe nonshort-circuitedbranch,andallthe

coincideatthe otherendofthisbranch.If

twogroups ofnodesseparately,oneineach

chesasthoughthey wereelasticbands,then

atsetof branches

lourhandsapart.

etchedbranches,the

uldbecutintotwo

ewouldbeholding

rpretationthat

t.'^

thusseento

minimumof

theactofcutting

rkintotwoparts.

tionofa cutsetas

onehandsomeof

andpullingtheseawayfromtherest(which

enedsomehowtotheplaneofthepaper);the

cutset.

mindlet usagainconsidertheformationof

seduponthegraph ofFig.5(a)withthe treeof

henceforthenode-pairvoltagesdefinedby

hisredrawnwiththe nodesletteredsowecan


/ h t t p : / / w w w . h




DNETWORKVARIABLES

Branches5,6,7,8 constitutethetree.Node-

nd thetipofits referencearrowisatnodeo.

pondingtothisnode-pairvoltage,theremaining

dasshortcircuits,whencethenodesb, c,d

enode o.Inourright handwe,therefore,

o,b,c,d, andwithourlefthand holdingnodea.

cearrowforeiis inourrighthand; hencethe

onforbranchesintheassociatedcutsetis from

and.Thestretchedbranchesclearlyarethose

rencearrowson4 and5areconfluentwith

ctionforthisnodepair whilethatonbranch1

tionofthefirstrowofschedule19 isthus

wweobservethate2 =v6,andsothe tipend

istime branches5,7,8are regardedasshort

edupin ourrighthandbecomeo,a,c,d, and

ctionforthecut-setbranchesisagaindivergent

t setconsistsofbranches1,2,6, with1and6

sareo, a,b,d,andfor e4theyareo,a, b,c.

derpickingupthatgroupofnodesthat coin-

pertinentnode-pairvoltage;thenthepositive

branchesintheassociatedcutset iscon-

epicked-upnodes.

fewadditionalremarksmaybeinorder

Thatisto say,ifwewereaskedto solvethese

msofthev's,the questionofuniquenessmay

equationsthan unknowns.However,the

ashasalreadybeendiscussedforthe current

s.12.Namely,amongEqs.20there are

ones(asmanyasthereare independentun-

elyamatterofseparatingfourindependent

roupandsolvingthese.The lastfourare

dyieldthedefinitionschosenforthee's in

ecanalternativelychoosesaythefirstthree

yields

5


/ h t t p : / / w w w . h




FCHOOSINGCURRENTVARIABLES23

inagreewith thedefinitions15.

ndent.Notany fourareindependent,but

urindependentonesinthisgroup,and there

ffour independentonesthatcanbefound

eforpickingfour independentonesisto

ngtothebranchvoltagesofapossibletree.

sthe expressionsforthee'sin termsofthev's;

olutionsforthe e'sintotheremainingequa-

discussedrelationsbetweenlinkvoltagesand

ecut-setschedulewhichcontainstheinforma-

ricalcharacterofthecutsets,as wellasthe

tweentheimpliednode-pairvoltagesandthe

entobe acompactandeffectivemodeof

doesfor theformulationofvariablesonthe

setscheduledoesforthe establishmentofa

currentbasis.Continueduse willbemadeof

he followingdiscussions.

hoosingCurrentVariables

ganappropriatesetofindependentcurrent

kproblemcanbe approachedinadifferent

imesbepreferred.Thus,themethodgivenin

inkcurrentswith asetofloop-current

schosenas loops,andtwopossibletrees.


/ h t t p : / / w w w . h




ANDNETWORKVARIABLES

tsor closedpathsuponwhichthesecurrents

fromthechoiceofa tree,whereasonemay

losedpathsfortheloopcurrentsat theoutset.

nthegraphofFig. 8.Inadditionto provid-

bersandreferencearrows,aset ofloopshave

gnatedwiththecirculatoryarrowsnumbered

identally,arereferredtoasmeshesbecause

eofthemeshesin afishnet.It isacommon

sistochoose,asaset ofcurrentvariables,the

tocirculateonthe contoursofthesemeshes.

e,wemustknowhowtorelate inanunam-

nner,thebranchcurrentstothechosenmesh

hroughsettingdownthetie-set schedule

cemadefortheclosedpathsdefiningthe tie

graphof Fig.8onehas,by inspection,

nsyield


/ h t t p : / / w w w . h





ningEqs.24,gives

anchcurrentsintermsofthe linkcurrents.

inFig.8,thebranches1, 5,8,9become

quationsingroup24,namely,

heexpressionsforthe meshcurrentsinterms

ese,the remainingEqs.24yieldagainthe

msofthelink currents,thus:

esultsexpressedbyEqs. 25and26are con-

Eqs. 27and28.Thatis tosay,thechoice^of

ththealgebraicrelationsbetweentheloop

rrents.Itmerelyservesasa convenientway

dentsubsetamongEqs.24.In thepresent

canjustaseasily pickanindependentsubset


/ h t t p : / / w w w . h




ANDNETWORKVARIABLES

concept;however,inmorecomplexproblems

seful.

shmentofaset ofcurrentvariablesthrough

oiceofclosedpaths,adifficultyarisesinthat

epathsisin generalnotassured.Anecessary

ditionisthatall branchesmustparticipatein

oneor moreofthebrancheswerenottraversed

urrentsin thesebranchesinadditiontothe

rtobeindependent.Actually,theloopcur-

rcouldnot beindependentsincealtogether

endentcurrents.

cessary)proceduretoinsuretheinde-

ths(tiesets) istoselectthemsuccessivelyin

ditionalpathinvolvesatleastnrift braiichthat

previouslyselectedpaths.Thisstatementfol-

pathsortiesets formanindependentset

edtie-setscheduleareindependent:thatis,

essanyrowin thisscheduleasalinear com-

f, aswewritedownthesuccessiverowsin

winvolvesabranchthathas notappearedin

hatrow cansurelynot

mbinationofthosealready

eindependentofthem.

owsthat thisprinciple

ofthefirst rowinvolves

secondrowintroduces

7,8;the thirdrow

andthelastrowinvolves

nch5.Itis notdifficult

onedesignatesonlymeshes

fcourse,possibleonly ina

a planeorsphere),then

dtie-setschedulecanalways

encethattheprinciplejust

simplechoicein aplane

e,alwaysassuresthein-

pathsandhencedoesthe

-currentvariables.

efor theIrowsin atie-setscheduletobe

llingthepropertyjustpointedout. Thus,

hispropertyofthe rowsisasufficientthough

oinsuretheirindependence.Whenclosed


/ h t t p : / / w w w . h





egeneralmanner,astheysometimesmaybe,it

outsetwhetherthechoicemadeisacceptable.

usreconsiderthenetworkgraphofFig.8

athsshownin Fig.9.Thetie-setschedule

theexpressionsforthebranchcurrentsin terms

+ i2+U

i~*2-U(30)

k=~*2- i3-U

inspectionofFig.9.

denceofthechosenloops,weobservethat

8indicatesthatthe branchcurrentsji,j2,k,k

Hencethefirstfour ofEqs.30shouldbe

uslyarenot,sincetheright-handmembers

uationsare identicalexceptforachangein

oopsindicatedinFig.9 arenotaninde-


/ h t t p : / / w w w . h




ANDNETWORKVARIABLES

ortie sets)isingeneralnot amatterthatis

oughonehasastraightforwardprocedurefor

n.Namely,thechosensetofloopsareinde-

eassociatedtie-setscheduleareindependent;

etofind inthisscheduleasubset ofIinde-

dependentequationsamongasetlike30).

rmakingthischeckamongthecolumnsisto

pondingtothelinksofanychosentree. These

I rowsofthescheduleareto beindependent.

quations(likethefirstfourof 30inthetest

aragraph)haveuniquesolutions.Usually

ectionwhetheror notsuchsolutionsexist.

odistosee ifthedeterminantoftheseequa-

nonvanishingofthedeterminantformed

scorrespondingtothelinks ofachosentree

endenceofanarbitrarilyselectedsetof closed

ngmanybranchesthismethodmayprove

tobeawareof alternativeproceduresfor

rrent-variabledefinitions,shouldthisbe

akeuseofthe factthatthemostgeneral

blethroughsuccessiveelementarytransforma-

venone,andthatsuchtransformationsleave

owsinvariant.Wemay,forexample,start

isbaseduponachoiceof meshessothatits

nt.Supposeweconstructanewfirstrow

entsofthe presentonetherespectiveelements

wscheduleis thenasshownin31.


/ h t t p : / / w w w . h





eshes2,3, 4ofFig.8.However,loop1

ourofmeshes1 and2,asa comparisonofthe

ulewiththegraphofFig.8 reveals.Ifwe

urtherbyconstructinganewsecondrowwith

of therespectiveonesofthepresentrows

notherschedulethatimpliesaloop 2withthe

shes2,3,and4.It shouldthusbeclearthat

setsarereadilyformedthroughcombining

mpleones.Solongasonly onenewrowis

binationofrowsinagivenschedule,andif the

stituentpartofthiscombination,theprocedure

ndenceofagivensetof rows.

epropertythatits columnscorrectlyyield

nchcurrentsin termsoftheimpliednewloop

cetransformationofthe schedulethrough

ationsimpliesarevisioninthe choiceofloops,

nin thealgebraicdefinitionsoftheloopcur-

ationsexpressingthebranchcurrentsin

rrentsis stillgivenbythe coefficientsinthe

orexample,wewouldgetforschedule31

+i'2

2(32)

g=-i'z

he i'stodistinguishthemfrom thoseinEqs.

chedule23.

32reveals thetransformationintheloop

nsformationofschedule23to theform31,

omewhatunexpected.Thusthetransforma-

hedule31 impliesleavingthecontoursforthe

ameasinthe graphofFig.8,but changesthe

Offhandwewouldexpectthealgebraic


/ h t t p : / / w w w . h




ANDNETWORKVARIABLES

eandthosefori2,%3,andi4to remainthesame.

3 thatt'i,^3,and%4 areunchangedwhiletj

s.32arecorrect,aswecanreadilyverify

the alteredcontourforloop1and expressing

arsuperpositionsofthe loopcurrents,noting

eresultsexpressedbyEqs.33,therefore,are

porarilycreatedbythisresultdisappears

ntionuponschedule23andEqs. 24andask

relations24willbring abouttheadditionof

23and leaverows2,3,and4 unchanged?

streplacethesymboli2byt'i +i2,forthen

alsoappearinrow1, inadditiontothe ele-

ow 1,andnothingelsewill change.Thelesson

ampleisthatweshouldnot expectasimpleand

eenthecontourschosenforloopcurrentsand

orthesecurrents,norshouldwe expecttobe

tionchangesinthe chosencontours(tiesets)

ormationsintheloopcurrentsuntilexperience

enus anadequateinsightintotherather

dbysuchtransformations.

misledinthefirstplacei sthatwearetoo

ofcontoursfor loopcurrentsasequivalentto

thebranchcurrents,whereasinrealitythe

erelyimpliesthealgebraicrelationshipsbe-

dbranchcurrents(throughfixingthe tie-set

ethemin evidence.

ear transformationofthetie-setschedule

ughwritingin placeof33

•+a2[i'i

d'1

alnumbers.Ift'i• ••ii areanindependent

ent'i •••t'j willbeindependentifEqs.34

they possessuniquesolutions(whichthey


/ h t t p : / / w w w . h





nonzero).Ingeneralthecurrentsi\• ••i'i

nificanceofcirculatorycurrentsorloopcur-

iencetheymaystillbereferredto bythat

besomelinearcombinationsofthebranch

ofdefinitionsfortheloop currentsisdesired,

structionofanappropriatetie-setschedule

ew,whichwewill illustrateforthenetwork

supposethatonewishes tointroducecurrent

owinglinearcombinationsofthebranch

9

heseexpressionsintermsof I(inthis case

othiswe mayfollowtheusualschemeof

he relationsforthetree-branchcurrentsin

or tree1ofFig.8, thesearegivenbyEqs.

qs.35 into

4

*4


/ h t t p : / / w w w . h




ANDNETWORKVARIABLES

e theadditionalrelations

)

38yieldtie-setschedule39,whichmore

ameinformation.Thisistheschedulethatis

rthe loop-currentvariables,whichnolonger

terpretationofbeingcirculatorycurrents.

wingchapter,thetie-setscheduleplaysan

lationofthe equilibriumequationsappropri-

nsforthe currentvariables.Thepresentdis-

dethebasisforaccommodatingsuchachoice,

ormodeof inception.Thuswehaveshown

ganappropriateset ofcurrentvariablescan

eedifferentforms:

hoiceofatree andidentificationofthe


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



FCHOOSINGVOLTAGEVARIABLES33

network),butnofacile controlishadregarding

efinitionsoftheloopcurrents.

makinganinitialand arbitrarilygeneral

finitionsofthecurrentvariables(likethose

asethevariablesnolonger possessthesimple

ofcirculatorycurrents.Thisapproachwill

andisgiven largelyforthesakeofits theo-

hoosingVoltageVariables

asvariables,wesimilarlyhavethreepossible

oftheapproachmaytake.Thefirst, whichis

dsthroughchoiceofa treeandtheidentifica-

eswithnode-pairvoltagevariables.Inthis

mentionedaboveforthechoiceofcurrentvari-

tionsforthenode-pairvoltagesareas simple

rnodirectcontrolcan beexercisedoverthe

fnodepairs.Asecondformofprocedure,

choiceofnodepairs

nwhichtheprocess

rarilygeneral choice

softhevoltagevari-

ndetail.

tionofnode-pair

pproachedthrough

ppropriatesetof

he networkofFig.8.

enodesofthisnet-

reaseof reference,

arrowheadsintended

epairs andreference

ariablesei,e2,e5.

confusedwith

yet,ifwemomentarily

noticethatthestructureinFig.10 hasthe

oritconnectsall ofthenodes,andinvolvesthe

hesneededtoaccomplishthisend.Hence

sei •••e5 isanappropriateonesincethe

dependentset,andtheirnumber equalsthe

treeassociatedwitha networkhaving

orthrightchoiceofnode pairsitis

hesystemofreferencearrowsaccompanying


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



DNETWORKVARIABLES

allydrawnormerelyimplied)formsastructure

ter.

thin Art.6,onecanconstructcut-set

othechoiceofnodepairs indicatedinFig.10


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e




41)

mayreadilybecheckedwithreferencetoFigs.8

nthatthev's aredropsandthee's arerises.

gedrop fromnodeatonode d.Ifwepass

ofnode-pairvoltagearrowsin Fig.10,we

rsethearrows fore2ande3counterfluently,

5,e^,andeiconfluently.Sinceconfluenceindi-

etermsforci,e4,and e5arenegative.There

us verifyingtheremainingequationsinset41.

s.41frominspectionofFigs. 8and10to

ctedschedule40by columns,whencethe

yieldthecutsets.Thispart oftheprocedure

easwith thealternateapproachgiveninArt.6.

gthesolutionofEqs.41 forthenode-pair

ranchvoltages.Oneselectsanyfiveindepend-

upand solvesthem.Againtheselectionofa

orkgraph(suchas tree1or2 inFig.8)is a

anindependentsubsetamongEqs.41,and

nyield theappropriateexpressionsforthe

e-branchvoltages,asdiscussedpreviously.

to theproblemofdefininganappropriate

evariables,arathercommonprocedureisto

earbitrarilyselectednodeas areferenceand

potentialsei. ••enof theremainingnodeswith

Thus,onenodeservesas adatumorreference,

the variablesei••• enallhavethis datum

ntitiesei•• •eninthis arrangementare

lsandare referredtoasa "node-to-datum"

eofnodepairsimpliedin thisspecialized

eparallelofchoosingmeshesforloopsin the

riables.Thisthemeiselaborateduponin

racteroftheloopandnodeproceduresisstressed

dualityare partiallyevaluated.


/ h t t p : / / w w w . h




ANDNETWORKVARIABLES

r achoiceofnode-pairvoltagesofthissort

enttothenetworkgraphofFig. 8.Again,for

arrowsinthis diagramasbranches,wesee

terandhence

msetofvoltages

one.

this groupof

ticularlyeasyto

settingall but

esequalto zero

coincideatthe

etip endof

ethebranches

odeformthe

rencetoFig.8,

readilyobtained.

esarethepo-

deswithrespect

branchvoltagedropisgiven bythe

entials,namelythoseassociatedwith

pertinentbranch.Ifthe lattertouchesthe

gedropis givenbyasinglenode potential

es


/ h t t p : / / w w w . h





eitherbyinspectionof Figs.8and11or

ule42to be

e4-e5

Ct>s=e2— e3(43)

=ei— e2

msof thebranchvoltagesarefoundfrom

of selectingfromtheseequationsasubsetof

cordingtotree1ofFig.8, thelastfiveare

onyields

nsinset 43thengivethefollowingexpressions

msofthetree-branchvoltages

vg

s+vg

ehowmoregeneralnode-pairvoltagedefini-

esimplenode-to-datumsetthroughcarrying

ontherowsof cut-setschedule42.Thus,

onea newschedulethroughaddingtheele-

42 totherespectiveonesofthe firstrow,


/ h t t p : / / w w w . h




ANDNETWORKVARIABLES


/ h t t p : / / w w w . h





e node-pairvoltagediagramhaschanged

11tothatshownin Fig.12,sincethepoten-

tothedatum(whichin Fig.11isez) nowis

e\.Wenotefurther that,whene\isthe

eseand/ coincideatthetipendof e'i;sothe

throughpickingup thesetwonodes,asisalso

hepicked-upnodescorrespondingtothe

gesevidentlyremainthesameasbefore,and

etsareunchanged.

e-pairvolt-Fio. 13.ThegraphofFig.8

orrespondingwithnodedesignationsas

setschedulegiveninFigs. 10,11,and12.

onsinschedule46maysimilarlybeinter-

w3is addedtorow4,the picked-upnodes

db,whichinFig. 12impliesthatthetail end

mtonodeb,and wewillfindthatnow63 ** e"s

merefers tothelatestrevisionof thesetof

stofthee's remainasinEqs.49 withdouble

uantities).

arryingoutadditionalrowcombinations

itisby nomeansalwayspossibletoassociate

mliketheonesin Figs.10,11,or12 withthe

es,forthereasonthatsomeof thesearelikely

entialdifferencesbetweennodepairsbut

nearcombinationsofthebranchvoltages.

nstructsacut-setschedule(asisalso apos-

garbitrarychoicesforthepicked-upnodes.

wemay consideragainthegraphofFig.8

withthenodesletteredasin Figs.10,11,


/ h t t p : / / w w w . h




DNETWORKVARIABLES

0is constructedbysimplymakinganarbi-

-upnodesrelatingtothe pertinentcutsets.

etainsonly anominalsignificancesincewe

theimpliedvoltagevariablesarepotential


/ h t t p : / / w w w . h





ngtree1in Fig.8designatesv5,v6,v7,vg,vg

chvoltagesandhencestipulatesthatthelast

ouldbeindependent.Itisreadily seenthat

solutions

cutsetsin schedule50areindependent,and

pliedvoltagevariablesarein termsofthe

woaresimplepotentialdifferencesbetween

hreearenot. Thereisnoreasonwhythe

shavetobepotentialdifferencesbetween

m anindependentset,andweknowthe

enthemandthebranchvoltages,theyare

e samenetworkofFig.8the followingset

binationsofthebranchvoltagesasa starting

,7+5vs+5v9

6+v7+4»s+4»9

»s+2f9

+2vs+vg

ecaneliminateallbutfiveof thebranchvolt-

into theform

f+5t'9

»9

4)


/ h t t p : / / w w w . h




ANDNETWORKVARIABLES

ationstogetherwithEqs. 45yieldthecomplete

nchvoltagesintermsofei •••eg, thus

marizedincut-setschedule56.Thus wesee


/ h t t p : / / w w w . h




proach.Wewishnowto callspecificattention

emsothatwemaygain thecircumspection

makeeffectiveuseofitsimplications.In a

sfromthefactthat twosituationswhich,on

srespectively,areentirelyanalogous,have

sexceptforaninterchangeoftherolesplayed

ilephysicallyandgeometricallytheyare

ycanonerecognizeanobviouseconomy

sultingfromthisfactsincethe analysisofonly

tedyields thebehaviorofboth,butonecan

erstandingoftheseideasmayleadto other

usefulapplications,asindeedthelaterdis-

stantiate.

viousarticlesin thischaptershowsthat

enceofideasandprocedurescharacterizes

emethods,butwithaninterchangeinpairs

ndconceptsinvolved.Sincethelatterare

alrole,theyarereferredto asdualquantities

gsuchdualquantitiesarecurrentandvoltage;

onceptsinvolvedaremeshesandnodesor

ea zerocurrentimpliesanopencircuitanda

,thesetwophysicalconstraintsareseen to

nofloopcurrentswithlink currentsandof

ee-branchvoltagesshowsthatthelinksand

aredualquantities.Theaccompanyingtable

of suchpairs.

ts

atdualityi sstrictlyamutualrelationship.

pairofquantitiesinthe tablecannotbe

achcolumnaswrittenassociatesthosequan-


/ h t t p : / / w w w . h




DNETWORKVARIABLES

epertinentto oneofthetwoprocedures

heloopandnode methodsofanalysis.

idtobe dualsifthecharacterizationofone

esultsidenticalinform withthoseobtained

theotheronthe nodebasis.Bothgraphswill

branches,butthe numberoftreebranchesin

nks intheother;or thenumberofinde-

equals thenumberofindependentloopsin

y,theequationsrelatingthebranchcurrents

etworkare identicalinformtotheequations

esandthenode-pairvoltagesforthe other,

tionsbecomeinterchangedifthe lettersiandj

y,byeandv,andviceversa.For appropriately

nchesoftheassociateddualnetworks,the

oftheseis obtainedfromthatoftheother

angeintheidentitiesof voltageandcurrent.

thatwillbe hadfromlaterapplicationsof

siderationoftheunderlyingprinciplesis

becauseoftheircorrelativevaluewithrespect

nsofthischapter.

saredualiftherelationshipbetween

one isidenticalwiththerelationshipbetween

ther.The detailedaspectsinvolvedinsuch

estseen fromactualexamples.Tothisend,

n Fig.14.Supposetheonein part(a)is

uctits dualasshowninpart (b).Atthe

graphof part(a)hassevenmeshesandfive

totalof sixnodes).Hencethedualgraph


/ h t t p : / / w w w . h




dentnodepairs(atotal ofeightnodes)andfive

ofbranchesmustbethe sameinbothgraphs.

structionofthedual of(a),onemaybegin

lcirclesas nodes—oneforeachmeshinthe

xtraonethatcanplay thepartofa datum

tassuch,althoughany ornoneoftheeight

redinthis light.Wenextassigneachof these

sevenmeshesinthegivengraph,as isindi-

e lettersa,b,• •.,g.Theproceduresofar

eringastie setsthoseconfluentbranchesin

ntoursofmeshesandascutsets thosebranches

stretchedintheprocessof pickingupsingle

ationistrue ofthenodesa,• ••,gthat are

es;thecutsetpertainingto theremaining

espondtoatie setingraph(a) thatwillreveal

carryouttheprocessofmakingall tiesets

to allthecutsetsin itsdual.

erencearrowsentirely;thesewill beadded

thmesha,weobservethat itspecifiesatie

1,6,7; thereforethecutsetformedthrough

ualgraphmustinvolvebranches1,6, 7,and

onfluentin nodea.Similarlythebranches

eshb,and thereforethesebranchesarecon-

graph;andso forth.Theactualprocessof

bestbegunbyinsertingonlythosebranches

otie setsandhencemustbecommontothe

to say,wenotethatanybranchesthatare

thegivengraph mustbecommontothetwo

edualgraphand hencearebranchesthatform

tweensuchnodepairs.Forexample,branch7

db, andhencebranch7inthe dualgraph

milarlybranch10 linksnodesbandc; branch

dsoforth.

ertbranches7,10,11,8,12, 9,andthennote

es1,2,3, 4,5,6in theoriginalgraphforma

calwiththecutset ofthedualgraphthat is

ningunassignednode.Hencethesebranches,

nanassignednode,are theonesthatmustbe

node.Thelatteris thusseentobeassignable

peripheryofthegivengraph.In asensewe

asa "referenceloop"correspondingtothe

eplayingtheroleofa "referencenode,"


/ h t t p : / / w w w . h




ANDNETWORKVARIABLES

cussionwillshowthatthisview isarather

otbeconsideredunlessit seemsdesirable

wsonthebranchesof thedualgraphwenote,

ersalofmesha inaclockwisedirectionis

cearrowofbranches1and6, andcounterfluent

fbranch7.Henceon thedualgraphwe

obranches1and6that aredivergentfromnode

h anarrowthatisconvergentuponthis node.

eclock-

eswith

ectivenodes,

owsin the

isagree

ngto

gbranch

gree or

sedirection

esh.We

consistent

lofthe

phchoose

ondingdirec-

elyre-

nthedual graph(whichwecandoanyway),

econsistentandsticktothe samechosen

eprocessofassigningbranchreferencearrows.

ctionofthegraphofFig.14(b),as thereader

ection.

atduality isinall respectsamutualrela-

ofindthatthe graph(a)ofFig.14 isrelatedto

detailedmannerthat(b),throughthe process

bed,isrelatedto(a). Thusweexpectthe

dto nodesin(a)asdo themeshesof(a)to

wefind uponinspectionthatsuchisnot

example,themeshingraph(b) havingits

secutivelytraversedbranches1,7,10,12,9,4

otto asinglenode,butinstead isseentobe

reenodessituatedat thevertexesofthe

nches2,3, 11,sincetheactofsimultaneously

ealsthesamegroup ofbranches1,7,10,12,

utset.

lly

b.


/ h t t p : / / w w w . h




cyiseasilyresolvedthroughconsideration

constructionofthedualofgraph (a)asshown

scorrespondtothenodesofgraph(a) inFig.

emeshesofgraph(a) correspondtonodesin

ereadershouldcarefullyverify.Theaddi-

ntheconstructionofthegraphofFig.15 is

hesaboutanynodeis chosentobeidentical

umberedbranchesaroundtherespective

entclockwise(orcounterclockwise)direction

hesandaroundnodes.Forexample,the

seorderaroundmeshaofthe graphofFig.

aroundnodeainthe graphofFig.15this

respondstocounterclockwiserotation.Corre-

esequenceofbranchesaroundmeshcinFig.

sisthe counterclockwisesequenceofthe

roundnodecinFig.15.Thiscorrespondence

esisseen toholdforall meshesandtheircorre-

etweenmeshesinFig.14(a) andnodesinFig.

eshesin Fig.15andtheircorrespondingnodes

tweenthesetwographs isindeedcomplete

sbetweenbranchcurrentsandloopcurrents

esandnode-pairvoltagesareconcerned,how-

mefor thegraphofFig.14(b)as theyarefor

ebothinvolvefundamentallythesamegeo-

eennodesandbranches,asacomparison

sonit isnotessentialin theconstructionofa

nch-numbersequencesaroundmeshesand

essonewishes forsomeotherreasontomake

againcorrespondtosinglenodesinthe original

toftheirelectricalbehavior,thenetworks

Figs.14(b) and15areentirelyidentical.

re,referredtoasbeing topologicallytequivalent,

rdedas thedualofFig.14(a),or thelatter

networksofFigs.14(b) and15.

xampleofdualgraphsis showninFig.16.

he graphofpart(a)correspondto similarly

wiserotationinonegraphwith counterclockwiserota-

rychoice.Onecanas wellchooseclockwiserotationin

beingthata consistentpatternisadheredto.

ctdealingwiththepropertiesoflinear graphsisknown


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



ANDNETWORKVARIABLES

ofpart(b);and,conversely,themeshesin

desin part(a).Itwill alsobeobservedthat

saroundmeshesandaroundcorresponding

restingtonoteinthis specialcasethat,although

of awheel,thespokesinone aretherimseg-

hs.

therusefultor ecognizethatthesegraphs

n Fig.17,wheretheytaketheformof so-

nswith"feedback"betweentheirinputand

k16inthe graphofFig.16(a)corresponds

the dualgraphofpart(b), sinceopen-and

Fig.16redrawninthe formofunbalancedladder

redualconcepts(aspreviouslymentioned).

onidentifiesthefirstnodeon theleftwiththe

elingbranches1and9 attheleftand branches

hladderconfigurationsaremuchusedin


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



ewellto knowthatthedualofa ladderis

sentialdifferencethatitsseriesbranchescorre-

the givenladder,andviceversa.

sofconstructingadualgraph,to visualize

ponthe surfaceofasphereinsteadofon a

theperipheryappearsasan ordinarymesh

ositesideofthe sphere.Forexample,ifthe

inedtoconsistofanelasticnet andisstretched

e untiltheperipherycontractsupontheoppo-

enowviewsthe spherefromtheoppositeside

shemisphere,thentheperipheryno longer

llydifferentincharacterfromanordinary

s asimpleopeninginthe net,likealltheother

es9,10,11,12, 13,14,15,16formingthe

armorelogicallyto correspondtothesimi-

anchesinthe dualgraph16(b)emanating

h,likeallthe othernodes,nowcorresponds

vengraph.

workvariables,oneidentifiesloopcurrents

e-pairvoltageswithtree-branchvoltages,it

e setconsistsofonelink andanumberoftree

etconsistsofonetree branchandanumberof

agivengraphcorrespondtocut setsinthe

sthatthetree branchesinoneofthesegraphs

tistosay, correspondingtreesindual

entarysetsofbranches.InFig.16,forexample,

es1,2, 3,4,5,6,7, 8ingraph(a) asforminga

ngtreeingraph (b)isformedbythe branches

6.Or,ifin graph(a)wechoosebranches1,2,

ingatree, theningraph(b) thecorresponding

5, 6,7,8,9,10, 11,16.

ordingtothe discussioninthepreceding

graphwepickatree andchoosethecomple-

sforminga treeinthedual graph,thenthe

enbranchcurrentsandloopcurrentsinone

dentical(exceptforareplacementoftheletters

ande)withthoserelatingbranchvoltagesand

ualgraph.In thegraphsofFig.16,for

branches1to8inclusiveas thetreeofgraph(a)

usiveasthetreeof graph(b).Theningraph

j\0,• ••,jiaare respectivelyidentifiedwith

>whilein graph(b)thebranchvoltagesvg,


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



DNETWORKVARIABLES

lyidentifiedwithnode-pairvoltagese\,

ranchcurrentsingraph(a) wethenhave,for

t+Jio; j3=-*2+*3=-Jio+jn,

gesin graph(b)wehavecorrespondingly

e2; t'3=—t'10+»n=-e2+e3, etc . The

seequationsasanexercise,andrepeatthe

eesaswell asforthegraphs ofFig.14.

hat similarresultsfora pairofdualgraphs

gevariablesareobtainedif foronegraph

opsandin theotherthecorrespondingnodesas

depairs.In thiscaseitmaybe desirableto

easa datumandthecorrespondingperipheral

ofadatummesh.Sincemoregeneralchoices

aybeexpressedaslinear combinationsofthese

ttheparallelismbetweenthecurrentand

etworksholdsinall cases,regardlessofthe

tingdefiningrelationsfornetworkvariables.

onotea restrictionwithregardtotheexist-

restrictionmaymosteasilybeunderstood

llpossiblechoicesoftie setsinagivennet-

utsetsin itsdual,andviceversa.In this

givengraphassomenetcoveringthesurface

sany confluentgroupofbranchesforminga

datthecloseof Art.5,letusthink ofinserting

athandthentyingoff,as wemightifthe

loon.Wewouldthusvirtually createtwo

heotheronlyat asinglepointwherethe con-

acommonnodefor thetwosubgraphsformed

balloons.Whetherwethus regardthetie

itsoriginalformuponthe sphere,itsprimary

presentargumentis concernedliesinthefact

longwhichthegivennetworkisdividedinto

inglythetotalityofmeshesis dividedinto

rrespondtotwogroupsofnodes.If we

se nodegroupsineachofour twohandsand

etchedbranchesplaceinevidencethecut set

tof theoriginalgraph.Theactof cutting

ltothetying-offprocessdescribedabove,since

aphisseparatedintotwoparts whichare,

wosubgraphscreatedbycontractingthetieset.

algraphandits dualdemandsthatto

neof thesetheremustcorrespondintheother


/ h t t p : / / w w w . h




ustdescribed.Itshouldbe clearthatthis

etifeithernetworkis notmappableupona

faceofsome multiplyconnectedspacelike

utora pretzel.Forexample,ifthemapping

rfaceofadoughnut,thenit isclearthata

ghtheholeis notatieset becausethedoughnut

rts throughthecontractionofthispath.

nectedregionlikethat ofasphereis theonly

pathsaretie sets.Thereisobviouslyno

ontheexistenceofcutsets,sincewecanvisu-

ntarygroupsofnodesinourtwohands and,

hedbranches,separatingthegraphintotwo

erthegeometrypermitsits beingmapped

phereisrevealedas anecessarycondition

graphshall correspondtoeverypossiblecut

helatteris constructibleonlyifthegraphof

ppable.*

gparagraphsofthe previousarticle,the

bjectofdualityis twofold.First,dualityisa

nalyticalequivalenceofpairsofphysically

asmappablenetworksareconcerned,it

ctoroftwothe totalityofdistinctnetwork

cur.Second,andnolessuseful,isthe result

gives ustwogeometricallydifferentwaysof

on;ifoneof theseprovesdifficulttocompre-

turnsoutto befarsimpler.Thischaracteris-

nterpretationsofdualsituationsto reinforce

prehendingthesignificanceofeitheronewe

ha fewtypicalexamples.

ablegraph,weconsideranode-to-datum

atisto say,wepickadatumnode,and

tentialsoftheremainingnodes withrespect

ishtoobtainalgebraicexpressionsforthese

likenumberofindependentbranchvoltages,

oselecta treeandrecognizethateachnode

ivenby analgebraicsumoftree-branch

onoftheseaswellas allforegoingprinciplespresented

hroughoutthesucceedingchaptersdealingwiththeirappli-

ortheconstructionofdualnetworksandthe evaluation

inthelastarticleof Ch.10.


/ h t t p : / / w w w . h




ANDNETWORKVARIABLES

omanynodeto thedatumviatreebranches

metricalpictureinvolvedandthepertinent

mpleandeasilycomprehensible.

pletelydualsituation.Fora givenmap-

hemeshcurrentsasa setofappropriate

gebraicexpressionsforthesein termsofalike

anchcurrents.Sincethelattermaybere-

asetoflinks associatedwithachosentree,

edureisclearlythesameas intheprevious

wever,thelucidityofthe pictureissuddenly

arto haveaprocedureforexpressingeach

aicsumof linkcurrentsthathasa geometric

dnesscomparabletotheprocessofexpressing

tree-branchvoltages,andyetwe feelcertain

tureofequivalentclaritysincetoevery

existsadualwhichpossessesall ofthesame

edegreeof lucidity.Ourfailuretofindthe

he oneinvolvingnodepotentialsmustbe

structinourmindsthecompletelydual

vethelatter,ourinitialobjectivewilleasily

andingofnetworkgeometrywillcorre-

etorecognizethedual geometrystemsfrom

whatismeantby amesh.Sinceweusethe

rticularkindof loop,namelythesimplest

ace,weestablishin ourmindstheviewthat

econtour(theassociatedtieset)insteadof

erto,namelythespacesurroundedbythat

ening—nottheboundaryofthatopening.

fanode—thepointofconfluenceofbranches.

nnectedbytreebranches.Thedual ofa

eforethedualofatree shouldbesomething

eshes)connectedbylinks.Ifweadd tothe

thesethoughtsthefactthattraversinga

crossingitatright anglesaregeometrically

anchvoltageisfound throughalongitudinal

abranchcurrentisgivenby asummationover

vewithoutfurtherdifficultyatthe geometrical

zedasthe dualofatree.It isthespace

tosectionsby thelinks.Eachofthese

e passesfrommeshtomeshby crossingthe

epassesfromnode tonodebyfollowingalong


/ h t t p : / / w w w . h




8showsin part(a)agraph intheformofa

t (b)apossibletreewith thelinksincluded

surroundingthetree, anddualtoit, isbest

zeasused todenoteafamiliarkindof picture

o traceacontinuouspathfromonepointin

outcrossinganyof thebarriersformedbythe

eshesmandn isshowndottedinpart (b)of

hepathleadingfrom onemeshtoanyother

hfromone nodetoanotheralongthetree

treeanditsdual whichisinterpretedasamaze.

apathsuch astheoneleadingfrommesh m

particularsetoflinks.Theselinkscharacterize

nfluenttree branchescharacterizethepath

n agiventree.

ualprocesses,wenowrealizethatwehave

eintheforegoingdiscussionswherewerefer to

altoanode-pairvoltage.Thelatter isthe

odepotentials,anditsdualis,therefore,the

eshcurrents,likethecurrentsinmeshesmand

nce(im— in)isalgebraicallygivenbythe

rrents(withdueattentionto sign)charac-

on,justas anode-pairvoltage(potential

odes)equalsthealgebraicsumoftree-branch

nnectingthisnodepair.The difference

calledamesh-paircurrent,isthereal dualofa

eadditionofthemazeconceptto our

geometry,wehaveacquiredageometrical

ofthealgebraicconnectionbetweenmesh-

kcurrentsthatisas lucidasthefamiliarone

entialdifferenceswithtree-branchvoltages.


/ h t t p : / / w w w . h




DNETWORKVARIABLES

clarifiedthroughmorespecificexamples.In

networkgraph[part(a)],itsdual [part(b)],and

hoiceofnode-to-datumvoltagescharacterizing

nthegraphofpart (a)thetreebranches

elinks(branches1,2,4, 5,6)areshowndotted.

),thesesamebranches(1, 2,4,5,6) formthe

s.Thedatumnodesurroundsthewholedual

),its dual(b),anda node-to-datumchoiceofnode-pair

gtothemeshcurrentsin(a). Thetreebranches(solid)in

n(b) andviceversa.

• •-,15are chosentocharacterizethegraph

ythenodepotentialse\,e2,.• •,e5characterize

h,it isevidentthattheexpressionsfor the

anchvoltagesread


/ h t t p : / / w w w . h




rrentsintermsofthelink currentsingraph

esultseither byexpressingthelinkcurrents

opcurrentsin thefollowingmanner,

ynotingthat eachmeshcurrent(likeanode

betweenthecurrentcirculatingonthecontour

mmeshcurrent,whichisvisualizedascircu-

theentiregraph.Inthis sensethedatum

tside thegraph,justasthe datumnodeinthe

lowingthepatternset inFig.18(b)for

differencesintermsof linkcurrents,onereadily

esentingthesituationdepictedin graph(a)

uslyrecognizeshowthealgebraicsignsin these

ereferencearrowsinvolved.

tworks,butwithanalteredchoicefor the

bles.InFig.20(a)are shownthepathsfor

dualgraph isnotrepeatedinthis figure,but

for thechoiceofnode-pairvoltagesinthe

dtothe newloopcurrentsingraph(a).All

othisrevisedchoicearedistinguishedbyprimes.

reisconcerned,onehaslittledifficultyin


/ h t t p : / / w w w . h




ANDNETWORKVARIABLES

as

respondingrelationsforthe loopcurrentsin

f thegraphinpart(a) ofFig.20mustbe

edthroughtheusual procedureofwriting

rentsin termsoftheloopcurrentsand solving.

ever,toestablishthem entirelybyanalogyto

forwelearn inthiswaymoreaboutthe

urrentsare relatedtothelinkcurrents.

3,forexample,surroundsthreemeshes,and

-pairvoltagee'3contributestothepotentials

Fig.20(b)].In formingthecutsetassociated

nodes0,p,q,whereasin formingthetieset

ysaythat we"pickup"themesheswhose

thattiesetinevidence.

tthatpickingupmeshesis dualtopicking

thatloopcurrents,ascontrastedto mesh

esultingcontoursofgroupsofmeshes,weare

node-pairvoltagediagram[likepart(b)of

chosenloop-currentdiagram[likepart(a)

xists,and,by analogytothedualvoltage

thepertinentrelationsfortheloop currents.


/ h t t p : / / w w w . h




ANDNETWORKVARIABLES

kthe indicatedmeshesasloopsandwritea corre-

.Selectanindependentsetofcolumnsasthosepertinent

e,andfromthecorre-

xpressionsforthemesh

msofbranchcurrents.

he treecomposedof

etreecomposedof

owthatthetwosetsof

sofj's areequivalent.

b) showthatthemesh

7,8asloop currents.

tofclosedpaths,and

e-setschedule.

ph ofProb.1,de-

heaccompanyingtie-set

pendentsetofloopcurrents.Ifso, expresstheloopcur-


/ h t t p : / / w w w . h




entsinlinks1,2, 3,4.Ineachcase,tracethe closedpaths

ents.

nches5,6, 7,8interms ofthelinkcurrents1, 2,3,4.

gtie-setscheduleanditsassociatedgraph,tracethe


/ h t t p : / / w w w . h




ANDNETWORKVARIABLES

ying,ineachcase,thenode-pairvoltageswiththetree-

choiceoftree,expressthelinkvoltagesin termsofthe

oltages.

andthedesignationof nodesshownintheaccompany-

datum node,andwriteacut-setscheduleforthe node-to-

. Expresstheseintermsof

tree-branchvoltagesspecifiedin

ph ofProb.3andthe node

6,determinewhichofthe

sare independent,andfor

tapertinentcut-setschedule,

voltagesintermsof thebranch

b.5:(a)02,04, 13,17,26,35,

46,57;(c) 02,06,13,15,24,

graphinProb.3,giving

mberingandreferencearrowsforall

phindicateaset ofmesh

o-datumvoltagesofProb.6,and showthatthecut-set

owtheappropriatetie-setschedule.Showfurtherthat

urrentsin termsoflinkcurrentsarei denticalinformwith

epotentialsin termsoftree-branchvoltagesfoundin

ob.8defineloop-currentvariables(mesh-paircurrents)

eindependentsetsofnode-pairvoltagesspecifiedinProb.7.

appropriatetie-set scheduleisidenticalwiththepertinent

,andthusfind therelationsbetweentheloopcurrentsand

ebranchvoltagesintree(a) ofProb.5.Foreachset of

s(makinguseoftheappropriatetie-setschedule)findthe

thsandtracethesein thedualgraph.

opriatelinearcombinations,showthatanysetoflin-

educibletothe particularsetshownhere,inwhichele-

eanyfinitevalues(includingzero).If necessary,some

respondingtochangesinbranchnumbering)are,of course,

e transformations.Thusshowthat,iftheIrows ofa

endent,itmustalwaysbepossibleto findatleastone set

dingtothechosentreeofanygivengraph arenumbered

ntsare definedasn=ji, it=jt,• ••,ii— ji,showthat

e-setschedulerepresentamatrixhavingl's onitsprin-


/ h t t p : / / w w w . h




olowerright) andallotherelementszero(calleda unit

ationwiththatinthe previousproblem.

onwithbranchesnumbered1to 5andadditional

achvertex(node)isconnectedwitheveryotherone.For

heduleprovethatanyfourofthe

areindependent.

raphof theprecedingproblem,

ie-setschedule,andprovethatany

either1 to5or 6to10 areinde-

hichabranchconnectseverynode

erminethenumbernof inde-

enumberI ofindependentloopsin

l nodesnt.Computethenumber

eededforthisgraphon theloopand

=2,3,4, 5,10,50,100,andtab-

nsionalgraphintheformof auni-

deson asideandn,' totalnodes.

dependentloopsis I=2(n,«— 1)

eshowingthenumbersnand I

wnhere,andchoosea treeconsist-

6inclusive.Lettheloop currentsbe

rk— 1•••5, andconstructtheper-

fineasecondsetofloop currentsas

ckwisedirectionaroundtheboundariesofthemeshes

tasecondtie-setscheduleappropriatetothischoiceof

erowsofthefirst schedulebythenumerals1• ••5and

thelettersa. .•e,expressthe rows(tiesets)ofeach

telinearcombinationsofrowsinthe otherschedule.For

. and5— e;4=d+e;etc .

elationshipsbetweenthetwosetsof closedpathsinvolved

rrents.Nowfindthealgebraicrelationshipsbetweenthe

%andthesetia••• i, ; thatistosay, expressthet'i••• it

viceversa.Comparethetopologicalandthealgebraic

ndnotecarefullythedistinctionthatmustbe madebe-

tuationin Prob.16,supposeweintroducesomenew

currentdifferencesgivenbythealgebraicrelationships

u— *z>=*,— tc

eduleappropriatetothesenewloopcurrents,andthus find

on whichtheycirculate(thatistosay,find thetopological

mpaniestheabovealgebraictransformation).

ctfromthesecondscheduleinProb. 16anewone

nations:b-a,c-a,d~b,e-c,e(parallelingtheabovealge-


/ h t t p : / / w w w . h




DNETWORKVARIABLES

dsketchinthegraphofProb.16 thesetofclosedpaths

edule.Denotingtheloopcurrentscirculatinguponthese

v, usethenewscheduletodeterminetheirvalues interms

dthusget theexpressionsfor*i• .•ivin termsofia. ..i,

ebraictransformationthataccompaniestheabovetopo-

ingthe graphofProb.16,thebranches1, 2,••. ,6

et ofrelayswhosedesiredoperationdependsuponthe

sbeingequalin pairsthus:ji— jt,jt= js,jt=j4, j4*■jt,

gascurrentvariablesthedifferences

=J3— J4, »<=J4— ]i, '5

pressedasa setofsimplernullconditionsn

leappropriatetothischoiceofcurrentvariables,and

tis anindependentset.Ifso,expressall ofthebranch

urrentvariables.

heduleshownwasobtainedfrom asetofmeshcur-

metricalnetwork,numberingallbranchesandincluding


/ h t t p : / / w w w . h




raphshownin theadjacentsketchitisproposedto

ltagesas

otentialsof theseparatenodeswithrespecttosome

tructthepertinentcut-setschedule,(b)Drawthetree

setschedule,(c)Aretherowsof thisscheduleindependent?

dicatesevenindependentcolumns.

helinesofintersectionof thethreemutuallyorthogonal

chotherandwiththesurfaceof aconcentricsphere,as

wmanyindependentvariablesareinvolved:(a)ona

tagebasis?Isit possibletochoosethecurrentsinthe

nindependentset?Showyour reasoning.


/ h t t p : / / w w w . h




iatesetofgeometricallyindependent

tageorthe currentbasis,oneisinterested

libriumofthe networkintermsofthese.

oingthisis givenbytheso-calledKirchhoff

o.Oneof theselawsexpressesafundamental

rmsofvoltages;theotherexpressesan analo-

urrents.Whencurrentsare chosenas

enetworkisexpressedbymeansof the

esarechosenasvariables,equilibriumisex-

.Thisseeminginconsistencywilladequately

ollowingparagraphs,butfirst letusbecome

hofflawsthemselves.

ussionofthevoltagelaw,andin preparation

ecallwhatismeantbyvoltage."Voltage"isa

ctricpotentialdifference."Electricpotential

osay,itis ascalarquantityliketemperature,

ude abovesealevel.Thefactthatit is

ctionofposition only(i.e.,asingle-valued

ngso farasKirchhoff'svoltagelawiscon-

koftheelectricpotentialof anypointina

epotentialofsomearbitrary pointchosen

nspeakof thealtitudeofanypointin amoun-

ttosealevel chosenasanarbitraryreference.

acterofeitherofthesefunctionsweimply

functionatsomepointrelativetothat at

heroutechosenin traversingfromoneof

nthecourseofactuallycarryingthrougha

ation.

twearetomeasurethe altitudeofthetip

ampshirewithrespectto somebenchmark

arymethodsusedinsurveying.Wedothis


/ h t t p : / / w w w . h




suringthedifferencesinaltitudebetweenap-

ediatepointsextendingoversomecircuitous

ectivelyatthe benchmarkandatthe mountain

asurementtobe negligible,thenetdifference

beindependentoftheroutechosen.A similar

se ofanyothersingle-valuedscalarfunction,

suchaone.

oon asurveyingtrip,startfromsomearbi-

ealloverthe mountainousterrain,andfinally

mark.In

o finda

de,as is

actthat

itare

onofthis

uselectri-

Kirchhoff's

teslightly.

etry

sewe pro-

,touching

,b, c,d,

odea.

us thepotentialofnodeb isthe"drop"in

evoltagedropv2 inbranch2.Similarlyv3

ch3andequalsthe potentialofnodeb

eedinginthiswayaroundtheperiphery,we

tionis true

iveto

slaws.

v 20+ v i6+ vg+ v i= 0

awexpressedforthe closedloopformedby

ythevaluesofsomeofthese voltagedrops

ve;otherwisethesumof allofthemcouldnot

asrepresentinganalgebraicsum.I fsome

mericallynegative,thenevidentlythepotential

f nodeb(bothreferredtothe samereference,

calmeaningofEq.1 isaidedbyuse ofthe

byregardingthenodes inthegraphof Fig.1

ntainousterrain,andthevoltagedropslike

betweenthepertinentbenchmarksinthe

s.Thusariseinaltitudein agivendirection


/ h t t p : / / w w w . h




TONS

scanalternatelyberegardedasanumerically

nyadditionalclosedpathsinthe network

ple,startingagainatnodeaandproceeding

d, e,and/,onemayreturn tonodeavia the

17,10,and4. Inthiscasethe voltage-law

optermisalgebraicallynegativewhentraversal

ontraryto itsreferencearrow.Thusthe

anchisjust whatitsnameimplies,namelyan

onwhichweagreetocallpositiveforthe volt-

evoltagedropactuallyhasthisdirection,its

ve;ifithas theoppositedirection,itsvalueis

osedcircuitwithintheterrain,the algebraic

ps(potentialdropsbetweenpertinentnode

freference-arrowdirections,asmustalsothe

eragivendrophas anumericallypositiveor

xample,is higherthanbenchmarkd,then

ve;and,sinceitsalgebraicsignin Eq.2isplus,

lvesanarithmeticsubtraction.Inbranch10,

tualdropinaltitudemaybecontraryto the

has anegativevalue.Thecorresponding

mericallypositive,asisappropriatesincewe

pinaltitudewhenweencounterbranch10in

ichEq.2 applies.

husexpressesthesimplyunderstandable

ofvoltagedrops inanyconfluentsetof

circuitorloopmustequal zero.Symbolically

bywriting

gmaisinterpretedas asummationsignand

esummedarevoltagedrops,with dueregard

ordisagreementoftheir pertinentreference

directionoftraversalaroundthe loop,thus

eplusor minussignrespectively.

eanimportantpropertyofequationsofthis

vennetworkgeometrysuchasthat shownin

+f20-v i7— v i0— v4=0


/ h t t p : / / w w w . h




oltage-lawequationsfortheupperleft-hand

andneighbor,thus:

tionsgives

=0(5)

equationpertinenttothe closedloopwhichis

eshescombined.Thereasonfor thisresultis

mmontoboth meshes,injectstheterms+vi

othetwoEqs.4, andhencecancelsoutintheir

suchcancelationofvoltagetermswill

onofanygroup ofequationsrelatingtomeshes

espondtobranchescommontothegroupof

eseparateequationsforthemeshesimme-

chEqs.4 refer,thus:

ave

4— v i3+v9=0(7)

o theperipheryoftheblockoffour upper

aphofFig. 1.Ifallthe equationsforthesep-

areadded,oneobtainsEq.1relating tothe

ph.Thestudentshouldtry thisasanexercise.

o ananalogouslawintermsof branchcur-

offcurrentlaw.Theelectriccurrentina branch

hargeflowsthroughthatbranch.Unlessthe

oragroup ofbranchesconfluentinthesame

ewillbe eithercreatedordestroyedatthat

aw,whichinessenceexpressestheprinciple

rge,statesthereforethatanalgebraicsumma-

fluentinthesame nodemustequalzero.

pressedbywriting(asin Eq.3):

supposewewriteequationsofthissort

oneimmediatelytotheright ofhinFig. 1.


/ h t t p : / / w w w . h




TONS

henetcurrentdivergingfrom apertinent

reeEqs. 9.Thisgives

a canceloutintheprocessof addition.

ig.1 revealsthatthesebranchesarecommon

esinquestion,whilethebranchesto which

q.10 referterminateonlyinoneof these

onmaybegiventheresultingEq. 10.

thegraphofFig. 1formedbybranches1,4,

sasubgraphofthe entirenetwork)asenclosed

essesthefactthatthe algebraicsumofcurrents

ualszero.In otherwords,thecurrentlaw

nga subgraphthesameasit doestoasingle

otpossibleforelectricchargeto pileupor

ainingalumpednetworkanymore thanitis

p ordiminishata singlenode.Thisfact

urrentlawappliedtoa groupofnodes,asshown

suallyhavedifficultyrecognizingthetruth

owfeelthatin aboxthereis moreroomfor

mayperhapsdo this,whereasatasingle

argewouldhavetojump offintospaceifmore

n anytimeinterval.Theaboveanalysis

tholdsforasimplenodemust holdalsofora

eKirchhoffLawEquations

asetof relationsthatuniquelydetermine

nymoment.Theymaybewritteninterms of

variables;theuniquenessrequirementde-

numberofindependentequationsshallequal

tvariablesinvolved.Wehaveseenearlier

kisexpressibleeitherintermsof I=b — nt+1


/ h t t p : / / w w w . h




HEKIRCHHOFFEQUATIONS

example,theloopcurrents)orinterms of

endentvoltages(forexamplethenode-pair

sisweshall,therefore,requireexactlyI

ndonavoltagebasisexactlynindependent

nour attentiontotheKirchhofflaws.It

owmanyindependentequationsofeachtype

urrent-lawtypes)maybewrittenforany

Considerfirstthevoltage-lawequations,and

eenwrittenforall oftheninemeshesof the

identally,thisgraphhas 20branchesand

0, nt=12).HenceI-20-12+1 =9,

berofmeshes.Anytree inthisnetwork

Thereare 9links,andhencethereare

entloopcurrents.

edoutin thepreviousarticle,itisclear

nwrittenforanyotherloop enclosingagroup

eformedbyaddingtogethertheseparate

tmeshes.Suchadditionalvoltage-law

dependent.Theinferenceisthatonecan

pendentequationsofthevoltage-lawtype.

edbythefollowingreasoning.Suppose,for

reeischosen,andthe linkcurrentsareidenti-

thecorrespondinglydeterminedloopsaset

rewritten.Theseequationsaresurelyinde-

esappearseparately,oneineachequation,

ossibletoexpressanyequationas alinear

Eachoftheseequationscouldbeused to

termsof tree-branchvoltages.Thisfact

swhatwassaidearlierwithregardtothe

ganindependentsetandthelinkvoltages

yintermsofthem(see Art.6,Ch.1).

forwhicha voltage-lawequationcouldbe

ormorelinks sincethetreebranchesalone

fin thisequationthepreviousexpressions

esare substituted,theresultantequation

entity 0=0,sinceno nontrivialrelation

chvoltagesalone(thetree-branchvoltages

earenotexpressiblein termsofeachother).

hevoltage-lawequationwrittenforthe

ressesnoindependentresult.Thereareindeed

age-lawequations.


/ h t t p : / / w w w . h




ONS

owtotheKirchhoffcurrent-lawequations

may beindependent.Referringagaintothe

begin writingequationsforseveralnodes

weexaminetheseequationscarefully,we

satleast onetermthatdoesnotappear inthe

considertheequationswrittenfor nodesa

rmsinvolvingj2 andj4donot appearinthe

at thej6andis termsintheequationfor

eonefor

equation

otheright

mswith

tained

fornodes

onsare

smani-

ssany

onofthe

stermsthat

rrent-

alnodes

tateof

nuestohold

eenwrittenfor allbutoneof thenodes.The

=nt— 1independentequationsofthe

ysbewritten.Thisconclusionis supported

.

kgeometry,atreeis chosen,andthetree-

fiedwithnode-pairvoltages.Forthecorre-

depairs,asetofKirchhoffcurrent-lawequa-

branchestakingpartintheequationfor

entcutset,just asthegroupofbranches in-

quationforanyloopis thetiesetfor thatloop.

henodepairdefinedbyany treebranch

ebranchinadditionto thoselinkshavingone

ponthe picked-upnodesJ(seeArt.8,Ch.1).

ce ofatreefor thenetworkgraphofFig.1,

odepair/,ejoinedbybranch 20,indicatesby

takepartinthepertinent cutset.Sincethe

dentifiedwiththerespectivenode-pair

ferencearrow pointingfrom/toe. That

ofFig. 1.

depair

20andthe


/ h t t p : / / w w w . h




PANDNODEBASES

odesaree,q,I,b, c,d.Hencethepertinent

s

Art.8ofCh.1 arehelpfulinwritingthe

achosensetofnodepairs, fortheelementsin

learethecoefficientsappropriatetothese

equationslike11arewrittenfor allofthe

tothen treebranches.Theseequationsare

etree-branchcurrentsappearseparately,one

certainlyisnotpossibletoexpressany equa-

onoftheothers.Eachoftheseequations

ne tree-branchcurrentintermsofthelink

tallysubstantiateswhatwassaidearlier

rentsbeinganindependentsetandthe tree-

ressibleuniquelyintermsofthem(see Art.5,

nentto anodepairforwhicha current-

tenwouldhavetoi nvolveoneormoretree

onnectsallofthenodes,and thereforenonode

tonetreebranchtouchingit.If insuchan

uationonesubstitutestheexpressionsalready

ree-branchcurrents,theresultantequation

entity 0=0,sinceno nontrivialrelation

ntsalone(thelink currentsareindependent

bleinterms ofeachother).Itfollows,there-

equationwrittenforanyadditionalnodepair

tresult.Thereareindeedexactlynindependent

sonthe LoopandNodeBases

tthatthestateof anetworkcanbecharac-

ermsof asetofI loopcurrentsorinterms of

ges,andhavingrecognizedthatthenumbersof

tage-lawandcurrent-lawequationsareI andn

onisimminentthattheequilibriumcondition

ssedin eitheroftwoways:(a)througha set

nwhichtheloop currentsarethevariables,

rrent-lawequationsinwhichthe node-pair

Theseprocedures,whicharereferredto


/ h t t p : / / w w w . h




TONS

dnodemethodsofexpressingnetworkequilib-

furtherdetail.

hod.Thevoltage-lawequations,likeEq.1,

voltagedrops.Iftheseequationsareto be

ntsas variables,wemustfindsomewayof

tagesintermsofthe loopcurrents.These

ntwosuccessivesteps.

latedtothe branchcurrentsbythevolt-

ingtothekindsofelements(inductance,

e)thatthebranchesrepresent;andthebranch

totheloopcurrentsin themannershownin

onoftherelationsbetweenbranchcurrents

rictedatpresentto networksinvolvingre-

teextensionstoincludetheconsiderationof

nceelementswillfollowinthe laterchapters.

ches1,2,3, •••be denotedbyti,r2,r3) etc.

nallthebranchvoltagesand allthebranch

orsettinguptheequilibriumequationson

ratedforthenetworkgraphshowninFig. 3.

raph,andpart(b)isa chosentree.Branches

kcurrentsji, j2,•••,ja areidentified

currentst'i,t'2,• •t'6.

duleisreadilyconstructedfromaninspection

spertinentto thesesixloopcurrents[asthe

ghplacingthelinks1,2,• ••,6,oneat atime,

TheKirchhoffvoltage-lawequationswritten

mediatelyobtainedthroughuseof thecoef-

• , b

ph(a),anda possibletree(b).


/ h t t p : / / w w w . h




PANDNODEBASES


/ h t t p : / / w w w . h




TONS

5/5,v+=37'4 volts,andsoforth.Useof

mequationsareobtainedthroughsub-

ev's intoEqs.14.Afterproperarrangement

methodofwritingequilibriumequationswe

nt-lawequations,likeEq.11 above,involve

seequationsareto bewrittenwiththe

ables,wemustexpressthebranchcurrents

voltages.Todothis,wenotethat thebranch

branchvoltagesthroughEqs. 12,andthe

relatedto thenode-pairvoltagesintheman-

ons12arenowmoreappropriatelywritten

19)


/ h t t p : / / w w w . h




PANDNODEBASES

eisthenreadilyconstructedfromaninspection

d-upnodespertinenttothesefournodepairs.

equationscorrespondingtothischoiceof


/ h t t p : / / w w w . h




TONS

125e3

o =0.111e4

umequationsareobtainedthroughsubstitut-

intoEqs.21.After properarrangement,the

e3+0.500e4= 0

3e3-0.750e4=0

e3+0.950e4= 0

e3+1.061e4= 0

servethattheprocedurefor settingup

olves,foreithertheloopor nodemethod,

ations:

sintermsof pertinentbranchquantities.

branchvoltagesandbranchcurrents.

ntermsof thedesiredvariables.

sandin thecolumnsoftheappropriatetie-set

themeansforwritingthe relations(a)and(c)

s(b),intheformof eitherEqs.12orEqs. 19,

case.

uationsareobtainedthroughsubstituting

e resultingonesinto(a).In theloopmethod,

evoltage-lawequations(a)are voltageswhile

are currents.Inthenodemethod,the

rrent-lawequations(a)arecurrentswhilethe

voltages.Therelations(b) areneededin

substitutionof(c) into(a);thatis tosay,

rsta conversionfrombranchcurrentsto

rsa.Itis thisconversionthatissuppliedby

enduponthe circuitelements(resistancesor

eexample).

duleisthus seentoplayadominantrole in

marizesincompactandreadilyusableform

eptthosedeterminedbytheelementvalues.


/ h t t p : / / w w w . h




duledefineanindependentsetofclosedpaths,

enientmeansforobtaininganindependentset

quations.Anyrowofacut-setschedule,on

sallof thebranchesterminatinginthesub-

ormorenodes.Sincethe algebraicsumof

anchesmustequalzero,therowsof acut-set

dea convenientmeansforobtaininganinde-

rrent-lawequations.

eschedulesprovidethepertinentrelations

variablesareintroduced.Theyareusefulnot

ainingtheappropriateequilibriumequations,

ablingoneto computeanyofthebranch

uesofthevariables.

metryisparticularlysimple,and where

orwarddefinitionsforthevariablesare ap-

cquiringsomeexperience,employamore

ningequilibriumequations(asgiveninArt. 6)

seofschedules.

eLoopandNode Bases

thefinalequilibriumEqs.18and 24are

thatthevariableii(resp. ei)appearsinthe

2(resp.e2) inthesecondcolumn,andsoforth.

orgranted,itbecomesevidentthattheessen-

byEqs.18,forexample,iscontainedwith

hincreasedcompactnessinthearrayof

nceparametermatrix.EquilibriumEqs.24are

thefollowingnode-conductanceparameter


/ h t t p : / / w w w . h




TONS

egiventoarectangulararrayofcoefficients

and26. Aswillbediscussedin laterchapters,

fsimultaneousalgebraicequationslikethose

ilemannerthroughuseof asetofsymbolic

lesofmatrixalgebra.Thesematters need

ent,however,sincethematrixconceptisat

achievetwoobjectivesthatcanbegrasped

matrixalgebrawhatever,namely:(a)torecog-

alinformationgivenbythesets ofEqs.18and

hencemoreeffectivelyplacedinevidence

rays25and26; (b)tomakeavailableagreatly

signatingloop-ornode-parametervaluesin

tivesmaybetterbeunderstoodthrough

commonsymbolicforminwhichequations

thus:

notedbyasymbollikern, r12,andsoforth.

reads

hismatrixis denotedbyr,* inwhichthe

endentlyassumeanyintegervaluesfrom1toI.

xdenotesthe rowposition,andthesecondone

onofthecoefficientwithrespecttoarray28.

equationslike24 wouldsymbolicallybe

0

n=0

=0


/ h t t p : / / w w w . h




RMATRICES

7in analyticform,withthespecific

ecessitate(withoutuseoftheparameter-

sandspace-consumingtaskcomparedwith

almatrix25.Useofthe matrixconcepttakes

therow andcolumnpositionofanumber

kvalue;itisno longernecessarytowrite

hosegivenby31.Similarremarksapplyto

onofparametersonthenodebasisand the

ndingparameter-matrixnotation.

yofParameterMatrices

5and26givenabovehavean important

commonwhichisdescribedastheir sym-

trix25wenotethat ri8=r2i= —18,

1=16,andso forth.Morespecifically,

sssymmetryaboutitsprincipaldiagonal,the

ytheelementsrn =35,r22= 19,r33=22,

dingfromtheupper left-tothelowerright-

ementssymmetricallylocatedaboveand

ual.Symbolicallythissymmetricalproperty

on

enode-conductancematrix26.

metermatrixisneither accidentalnor

opertyoflinear networks.Itistheresult

rateprocedureinthederivationof equilibrium

meansalwaysbe adheredto.

natureofthisprocedure,letus recallfirst

gequilibriumequationsinvolvespredom-

17,


/ h t t p : / / w w w . h




TONS

ationsdesignatedinthesummaryinArt.3

uationsand(c)the definingequationsforthe

cuitelementrelations(b)areneededincarry-

(c)into(a)but arenotpertinenttothe present

sisthevariablesare loopcurrents,andthe

thevoltage-lawtype;onthenodebasis the

tages,andtheKirchhoffequationsareofthe

p-currentvariablesinvolvesthefixingofa

s(tiesets),eitherthroughthe choiceofatree

kcurrentswithloop currentsorthroughthe

of geometricallyindependentloops.The

e-lawequationsalsonecessitatestheselection

ndependentloops,butthissetneednotbe the

he definitionofthechosenloopcurrents.If

thedefinitionofloopcurrentsandi nthe

equations,thentheresultingparameter

trical,butifseparatechoicesaremade forthe

currentsandthosefor whichthevoltage-

thentheparametermatriceswillnot become

dureforobtainingthe loopequilibrium

eoftwotie-setschedules.Oneofthese per-

setofloop-currentvariables(asdiscussedin

ntheother oneservemerelyasabasis for

uations.Insteadofusingtherowsand col-

eforobtainingrelations(a) and(c)respectively

above,oneuses therowsofonescheduleand

ereadershouldillustratethese mattersfor

hthisrevisedprocedureforthe numerical

notingthedetailedchangesthatoccur.

basis,onemustchooseaset ofgeometrically

dtheirassociatedcutsetsfor thedefinitionof

es,andagainforthewritingof theKirchhoff

esecondselectionofnodepairsandassociated

ameasthefirst, but,iftheyare(as inthe

gtoEqs.24),thentheresulting parameter

rical.

dureforobtainingthe nodeequilibrium

eoftwocut-setschedules.Oneoftheseper-

setofnode-pairvoltagevariables(as dis-

ecutsetsin theotheroneareutilized inwriting


/ h t t p : / / w w w . h




steadofusingtherowsandcolumnsofthe

herowsofone scheduleandthecolumnsof

sethoughtsis thatthechoiceofvariables,

e,needhavenorelationto theprocessofwrit-

ns.Itismerelynecessarythatthelatter bean

blesintermsof whichtheyareultimately

nwithcompletefreedom.

eusedforvoltage-lawequationsandloop-

amecutsets areusedforcurrent-lawequa-

edefinitions,thenwesaythat thechoiceof/

htheKirchhoff-lawequations.Itisthis con-,

metricalparametermatrices.*

intheparametermatricesis important

recognizethedeliberatenessintheachieve-

t(asis quitecommon)becomeconfusedinto

entpropertyoflinearpassivebilateralnet-

bysymmetricalparametermatrices.We

usualprocedurethatleadstosymmetry,not

wochoicesbeingmadefora setofloopsor

sesymmetricalequationsareeasiertosolve,

nterestingnetworkpropertiesaremorereadily

dwefollowthe customaryprocedure,but

spectivethatcomesfroma deeperunder-

nvolved.

atAreAdequateinManyPractical

ngverygeneralapproachtothe matterof

uationsofnetworksbecause,throughitasa

napositionto understandfarmoreadequately

tisfactionthefollowingratherrestrictedbut

ceduresapplicabletomanygeometricalnet-

withinpractice.Thus,inmanysituations

pointedoutby theauthorataninformalround-table

alysisandsynthesissponsoredbytheAIEEatits midwinter

scussions(supplementedbyadistributionofpertinent

ludedderivationofthegeneralloopandnodeequilibrium

worksinsymmetricalordissymmetricalformandthecon-

ningsymmetricalmatricesfornetworkscontaininguni-

nappropriatedefinitionofvariables.Duringthepast 15

hismaterialwascontinuallysimplifiedthroughclassroom


/ h t t p : / / w w w . h




TONS

ngwork,thenetworkgeometryissuchthatthe

planesurfacewithouthavinganybranches

onedinArt.9,Ch. 1suchanetworkis spoken

plane,"ormorebrieflyasa mappablenet-

graphisshowninFig. 3isnotof themap-

ivenby thegraphinFig.1 is.

ationsforamappablenetwork(suchas

obewrittenon theloopbasis,itis possible

lyindependentsetofclosedloopsthe meshes

ointedout inArt.7of Ch.1).Asimple

wninFig.4in whichthemeshesareindicated

correspondingvoltage-lawequationsare

msofthe loopcurrentsareseentobe givenby

ancevaluesare

=2, r5=10, r6=5(35)

spectivelybythesevaluesyieldthecorre-

hichEqs.33 becomeexpressedintermsof

operarrangementthissubstitutionyields

0

rix


/ h t t p : / / w w w . h




tationmaybegiventothese equationsby

the samenetworkasinFig.4 isredrawnwith

referencearrowsleftoffbutwith thebranch

esindicated.Theterra7*iin thefirstofEqs.

e voltagedropcausedinmesh1by loop

esistanceonthecontourofthismeshis 7ohms;

sequationrepresentadditionalvoltagedrops

oopcurrentsi2,13,U, respectively.Sinceno

sh1istraversedbythe currentsi2andi3,these

kgraphFig.5.The resistancenetworkwhose

osenas graphisshowninFig. 4.Element

nmesh 1;hencethecoefficientsoftheirterms

ero.Theterm —2i4takesaccountofthefact

versingthe2-ohmresistance,contributesto

andthatthiscontributionis negativewith

cearrowin mesh1.

arlyexpressesthe factthatthealgebraic

edinmesh 2bythevariousloop currents

mshavenonzerocoefficientswhoseassociated

astpart ofthecontourofmesh2.The value

equalstheohmicvalueofthetotal orpartial

edbythepertinentloop current,anditsalge-

accordingtowhetherthereferencedirection

or disagrees,respectively,withthereference

usremarksapplytothe restofEqs.36.

mind,one canwritetheloop-resistance

ecoefficientsontheprincipaldiagonalare,

stancevaluesonthecontoursofmeshes1,2,


/ h t t p : / / w w w . h




TONS

efficientsareresistancesofbranchescommon

heiralgebraicsignsplusor minusaccording

erfluenceoftherespectivemesharrowsinthe

h.Specifically,atermr,kin valueequalsthe

ommontomeshessand k;itsalgebraicsignis

vethesamedirectioninthis commonbranch;

positedirections.

ththemesheschosenasloops andtheloop

entlyclockwise(orconsistentlycounterclock-

ofall nondiagonaltermsintheloop-resistance

bviousthat thisprocedureforthederivation

onsyieldsasymmetricalparametermatrix

ommontomeshess andfc,whosevaluedeter-

satthe sametimecommontomesheskands.

orwritingdowntheloopequilibriumequa-

eachoiceforthe loopsandloopcurrents)

emappabilityofthenetwork,butit isnot

tsoonloses itssimplicityanddirectness

trybecomesrandom.For,inarandomcase

ontinuetospeakof meshesassimplified

er,theirchoiceiscertainlynolongerstraight-

onof loopreferencearrowsassimpletoindi-

aybecommontomorethantwomeshes;

cearrowsmaytraversesuchabranchin

thenondiagonalcoefficientsintheparameter

onsistentlynegative.Althoughthesimplified

blein somemoderatelycomplexnonmappable

egeneral proceduredescribedearlierprefer-

rkgeometriesareencountered.

ocedureappropriatetorelativelysimple

forthedeterminationofnodeequilibrium

dprocedurethenode-pairvoltagevariables

atumset,asdescribedin Art.8ofCh.1. That

potentialsofthevarioussinglenodes with

trarilyselected)datumnode,asillustratedin

workgraphofFig.8. Thecutsets(which

urrent-lawequations)arethenallgivenby

ergentfromthesingle nodesforwhichthe

aredefined.

of Fig.4onemaychoosethebottomnode

anddefine thepotentialsofnodes1and 2

evariableseiande2.Noting thatthepertinent


/ h t t p : / / w w w . h




vergentfromthese nodes,thecurrent-law

thisselectionofnode-pairvoltagesareseen

msofthe nodepotentialsare,byinspection

scorrespondingtotheresistancevalues35 are

5,ff4 =0.5,g5=0.1, g6=0.2

spectivelybythesevaluesyieldthecorre-

enodepotentials.Theirsubstitutioninto

edequilibriumequations,whichread

e-conductancematrix

tationmaybegiventothe nodeequilibrium

nterpretationgivenabovefortheloopequa-

nthefirstof Eqs.41representsthecurrent

om node1bythepotentialei actingalone

secondterminthis equationrepresentsthe

vergefrom node1bythe potentiale2acting

).Sinceapositivee2actingalone causes

node1(insteadof causingadivergenceof

s numericallynegative.Theamountofcur-

divergefromnode1evidentlyequalsthe

conductancebetweennode1anddatumwhen

2coincideswiththe datum).Thistotal

esumoftheconductancesofthevarious

ode1;withreferencetoFig.5 (inwhichthe


/ h t t p : / / w w w . h




TONS

eresistances)thistotalconductanceis1/5

husaccountingforthe coefficientofthe

qs.41.

ausesto divergefromnode1cantraverse

tingnode1directlywithnode2 (thesearethe

esinFig.5),and thevalueofthiscurrentis

debytheproductofe2 andthenetcon-

edbranches.Inthepresentexampletheper-

+1/5=0.70 mho,thusaccountingforthe

hesecondtermof thefirstofEqs.41 (the

has alreadybeenexplained).Asimilar

ventothe secondofEqs.41.

eirconductancematrix42couldbewritten

nofFig.5,especiallyifthe branch-resistance

enasbranch-conductancevaluesexpressed

he principaldiagonalof[G]are,respectively,

ues(sumsofbranchconductances)divergent

moregeneralcasetherewillbemore thantwo

elementsof[G]allhavenegativealgebraic

enabovein thedetailedexplanationofEqs.41

oallcasesin whichthenode-pairvoltage

ode-to-datumset.Inmagnitude,thenon-

qualthe netconductancevalues(sumsof

thosebranchesdirectlyconnectingtheper-

ecifically,theelementg,kin[6] equalsthe

ctancesofthevariousbranchesdirectlycon-

hesenodesarenotdirectlyconnectedbyany

entg,kvalueiszero.Note thattheconsistent

agonaltermsfollowsdirectlyfromthetacit

potentialisregardedas positivewhenitis

tumnode.Thissituationparallelsthecon-

enondiagonaltermsinthe[R] matrixobtained

ppablenetworkinwhichallthemesh reference

entlyclockwise(orconsistentlycounterclock-

monbranchtheyarecounterfluent.

companyingvoltagedropsexistinare-

beingdissipated.Sinceateveryinstantthe

equalits rateofdissipation,therecanbe

purelyresistiveorinany "lossy"network

eor moresourcesofenergy.


/ h t t p : / / w w w . h




y sourceshasnotbeenintroducedintothe

dtheirpresencehasnothingwhatevertodo

ofar.Sourceswerepurposelyleft outof

on,sincetheirinclusionwouldmerelyhave

venessofthediscussion.Now,however,it

gnificanceofsources,theircharacteristics,and

eir effectupontheequilibriumequations.

ct,asalreadystated,isthat withoutthem

e.Thisfactmay clearlybeseen,forexample,

qs.36 forthenetworkofFig.5.Since these

efourunknownsi2,iz,u areindependent,

membersarezero,weknowaccordingtothe

ut thetrivialsolutiont'i= *2=iz= U=0

eabsenceofexcitation(which,asweshall

membersoftheequationstobe nonzero)

dasadoornail."

troductionthatanelectricalnetworkaswe

ithourpresentdiscussionsisalmostalwaysan

somephysicalsystemintermsofidealized

ecircuitelementsorparameters(theresist-

pacitanceelements).Wejustifysuchanarti-

ghnoting(a)thatit canbesochosenasto

toanydesireddegreeofaccuracy)theactual

ntsofinterest,and (b)thatsuchanidealiza-

gtheanalysisproceduretoarelativelysimple

eform.

oughwhichthenetworkbecomesenergized

calsystemderivesits motivepower,acon-

onisnecessary.Thatis tosay,thesources,

rerepresentedinan idealizedfashion.We

ysourcesmaythus besimulatedthroughsuch

nationwithidealizedcircuitelements.For

ttentionuponthei dealizedsourcesthemselves.

ctionofasourceisto supplyenergytothe

e beingfinditmoreexpedientto characterize

pableof providingafixedamountofvoltage

entatacertainpoint.Actuallyitprovides

andhenceanamountofpower equaltotheir

yessentialandpracticallymorerealistic

voltageorthecurrentofthe sourceisknown

se,postulateasourcefor whichboththe

efixed,but suchsourceswouldnotproveuseful


/ h t t p : / / w w w . h




TONS

alsystems,andwemust atalltimesbe mind-

hodsofanalysis.

geor thecurrentofasourceis fixed,wedo

tis aconstant,butratherthatits value

continuousfunctionofthetimeareindependent

rrentsin theentirenetwork.Mostimportant

ondependenceuponthesource'sownvoltage,

owncurrentifitis avoltage.Thusaso-

urceprovidesatagiventerminalpair avoltage

ntofthecurrentat thatterminalpair;andan

rovidesacurrentfunctionthatisi ndependent

nentterminalpair.

efultocomparetheidealizedsourceasjust

assiveresistanceorothercircuitelement.

ndcurrentattheterminalsare relatedina

the"volt-ampererelationship"forthatele-

sistancethevoltageis proportionaltothe

oportionalitybeingwhatwecallthevalue

theterminalsofanidealvoltagesource,on

eis whateverweassumeittobe,and itcannot

pecification,regardlessofthecurrentitis

countofthe conditionsimposedbyits

esituationarisesiftheenvironmentisa short

s calledupontodeliveraninfinitecurrent;

yandwithoutitsterminalvoltagedeparting

signedvalue.Itis,ofcourse,not sensibleto

ceinsucha situation,foritthenis called

er.Theideal voltagesourceisidlewhen

circuit,for thentheassociatedcurrentbe-

ofan idealcurrentsourcethecurrentis

be,and itcannotdepartfromthisspecification,

s calledupontoproduceonaccountofthe

environment.Anextremesituationarises

mentturnsouttobe anopencircuit,forthen

ninfinitevoltageat itsterminalssincethe

tion,cannotdepartfromitsspecifiedvalue.

ltagesource,itiscalleduponto deliverin-

s notrealistictoplaceanideal currentsource

ment.Thistypeofsourceisidle whenshort-

atedvoltageisthenzero.

off'svoltagelawwefound itusefultothink

altitudein amountainousterrain.The


/ h t t p : / / w w w . h




sinthe networkwithrespecttoacommon

oughtofas beinganalogoustothealtitudesof

nousterrainwith respecttosealevelasa

adofanactualmountainousterrain,suppose

plicaconstructedbyhangingupa large

ngfromit variousweightsattachedatrandom

eanalogueofvoltage,theproblemof finding

tionsonthe sheet(above,say,thefloorasa

logoustodeterminingthepotentialsofvarious

orkwithreferencetoa datumnode.

dertheelectricalnetworktohaveno sources

ntialsarezero.Theanalogoussituationin-

ouldbeto haveitlyingflaton thefloor.

ontothe networkmayberegardedascausing

lsto begivenfixedvalues.Analogously,

sheetareraisedabovethefloorto fixed

re.Asaresult,the variousnodesinthe

potentialsarenotarbitrarilyfixed,assume

entwiththeappliedexcitationandthe char-

Analogously,thefreelymovableportions

epositionsabovethe floorlevelthatare

whichthesheet issupportedatthepoints

goustoexcitationofthe electricalnetwork)

eristicsofthesheetwithits systemofattached

mthedescriptionofthesetwoanalogous

xcitationbymeansofvoltagesourcesmaybe

ngor clampingthevoltageatacertain

ourceis thusregardedasanappliedcon-

bersheet tothewallat somepoint.

nusedtoexcitean electricnetworkmay

pliedconstraints.Inanypassivenetwork

nits variouspartsareingeneralfree toassume

onlyto certaininterrelationshipsdictatedby

rk,but,withoutanyexcitation,allvoltages

fwenow givetosomeofthesevoltagesand

ovalues,wetakeawaytheirfreedom,forthey

valuesexceptthespecifiedones,but there-

ents,whosevaluesarenotpegged,nowmove

patiblewiththenetworkcharacteristicsinter-

rrents,andwiththe fixedvaluesofthose

excitationquantities.Asmoreofthevoltages

orfixedthroughthe applicationofsources,


/ h t t p : / / w w w . h




TONS

tthemselvestocompatiblevalues.Finally,

wereconstrainedbyapplied sources,there

emleft,foreverythingwouldbe known

onestsituation,onlyasinglevoltageorcur-

dthroughanappliedsource;determination

fallthe othersconstitutesthenetworkproblem.

rcesareschematicallyrepresentedincircuit

6.Parts (a),(b),and(c)are representations

aspart(d)showstherepresentationfora cur-

a)and(b)arecommonwaysofindicating

ntationsforsources,(a)Aconstantvoltage(batiery),(b)

nerator),(c)arbitraryvoltagefunction,(d)arbitrarycur-

alsocalled"directcurrent"or"d-c"voltage

)simulatesabattery,forexample,adrycell

(thinline) isnegativeandthecarbonelec-

sitiveterminal.Thed-csourceshownin(b)

ommutatorandbrushesof agenerator.The

n(c)isintendedtobe moregeneralinthatthe

ndicatesthate,(t) maybeanyfunctionof

usoid,althoughthereis anestablishedprac-

stherepresentationforasinusoidalgenerator).

tedthat e,(l)inthesymbolicrepresentation

efunctionand,inparticular,mayalsobe

voltagesource(d-csource).

eschematicrepresentationforacurrent

ytimefunctionandhencemaybeusedto denote

wellasanyother.

esentationsitwillbenotedthat areference

wdoesnot implythatthesourcevoltageor

theindicateddirectionbutonlythat,ifit

ethisdirection,it willatthatmomentbe

ntity.Thereferencearrowestablishesa

quantitye,(t)or ispositiveandwhenit is


/ h t t p : / / w w w . h




eissaidto "actinthedirectionofthe refer-

oltageriseinthisdirection.The+ and—

fFig.6 furtherclarifythisstatement.In

therepresentationsshowninparts (c)and

edthattherepresentationsinFig.6are for

tagebetweentheterminalsinthe sketch

omatterwhatisplacedacrossthem.Like-

mthe terminalsinthesketchofpart (d)is

ttheexternalcircuitmaybe. Anactual

ay,toafirstapproximation,berepresented

eries withtheidealoneso thattheterminal

ourcecurrentincreases.Aphysicalcurrent

presentedtoafirst approximationthrough

tharesistanceinparallel withtheterminals,

factthatthe netcurrentissuingfromthe

ondependsupontheterminalvoltage,and

ncreases.Thesematterswillfurtherbe

licationstocomelateron.

ntsthattheyhave moredifficultyvisualiz-

canceofcurrentsourcesthantheydointhe

sources.Acontributingreasonforthis

urcesaremorecommonlyexperienced.Thus

pplyelectricitytoour homesandfactories

rcesinthattheyhavethe propertyofbeing

Sourcesthatarebasicallyofthecurrent

n.Onesuchsourceisthe photoelectriccell

rtionaltotheintensityofthe impinginglight

rrentsource;it clearlyisidlewhenshort-

eliversnoenergy.Anotherdevicethatis

urrentsourceisthepentodevacuumtube.

arlyproportionaltoitsgrid excitationunder

ns,andhence,forpurposesofcircuitanalysis,

eritasbeingessentiallya currentsource.

ygoodaccuracyberegardedasanideal cur-

a resistance.

emorecorrectlytoberegardedas voltage

esis,however,arather pointlessargument

t eitherrepresentation(incombinationwith

entofpassivecircuitelements)isalwayspos-

tualsourcereally is.Againwemustbe

rymakesnoclaimtobe dealingwithactual


/ h t t p : / / w w w . h




TONS

itelydealsonlywithfictitiousthings,but

hingscantherebyberepresented."Likeall

circuittheoryismerelythe meanstoanend;

erealthing.

wsourcequantitiesenterinto theequilib-

network,wefirst maketherathergeneral

ionofsourcesintoa givenpassivenetwork

ys.Oneoftheseis toinsertthesourceinto

vingvoltagesource(constraint)inparallelwitha branch

sedgraph(b) showingdispositionofvoltagesource.

a branch(aswithapliers);the otheristo

alstoaselectednodepair (aswithasoldering

willbedistinguishedas the"pliersmethod"

ethod"respectively.Weshallnowshowthat

smethodrestrictedtothe insertionofvoltage

ironmethodtotheinsertionof currentsources.

tionofavoltagesourceacrossanodepairor

sourceinserieswitha branchimpliesarevision

withtheendresult thatvoltagesourcesagain

ranchesand currentsourcesappearonlyin

crossnodepairs).

Fig.7isshowna graphinwhichavoltage

elwithbranch6 ofsomenetwork,andinpart

heresultantchangein thenetworkgeometry

whichthissituationreducesto.Thus,in

ngementinpart(a),oneshould firstobserve

trivialbyhavinge,placedin parallelwithit


/ h t t p : / / w w w . h




usforcedtobeequalto e,andhence(along

known.Thatistosay,the determinationof

enderedtriviallysimpleandindependentof

ofthenetwork.Thereforewecanremove

sand fromtherestofthe graphsothate,

tinglinkbetweennodesaand b.Nextweob-

nodesc, d,f,relativetothat ofnodeaare

rrangementofpart (b)inFig.7 astheyare

hepotentialofnodecwithrespectto that

evidentby inspectionofeitherpart(a)or

larlythepotentialofnoded withrespect

be (e,+1^7)inthe arrangementofpart(a)

mesclearthatthe branchvoltagesandcur-

b)mustbe thesameasinthe graphofpart

nofthe trivialbranch6.

cingavoltagesourceacrossanode pair

henetworkgeometryasplacinga shortcircuit

paringgraphs(a)and(b) inFig.7,wesee,

gesourcee,in graph(a)effectivelyunites

h,thuseliminatingbranch6, andyielding

effectof thevoltagesourcesofaras this

distakenintoaccountthroughplacingiden-

eswithall branchesconfluentintheoriginal

yplacetheidenticalvoltagesourcesin series

yconfluentinnodea:that is,inbranches

nd9.

ontoregardavoltagesourceasthoughit

dshortcircuit,whichindeeditis.Thus,bya

orbranchforwhichthepotentialdifference

roindependentofthe branchcurrent,while

tentialdifferenceis e,independentofthe

the shortcircuitisidenticalwiththe voltage

ta deadvoltagesourceisashort circuit.

showsthattheeffectofa voltagesourceupon

esameas thatofanappliedshort-circuit

.8depictsa situationinwhichacurrent

with branch4ofsomenetwork,andpart(b)

eingeometryandsourcearrangementwhich

erencetothegiven situationinpart(a)

ranch4becomestrivialsinceits currentis

urrentandhenceis known.Itisalsoevident


/ h t t p : / / w w w . h




TONS

ntsourcei, upontherestofthe networkis

ad beennobranchlinkingnodesa andb

sapplied.Wecan,therefore,regardthe cur-

cross thenodepaira-bin amodifiedgraph

nt.

inhavingallcurrentsourcesin parallel

iedoutas showninpart(b)of Fig.8.The

nticalcurrentsourcesi,bridgedacross

vingcurrentsource(constraint)inserieswitha branch

sedgraph(b) showingdispositionofcurrentsource.

asinglesourcei,bridgedacrossthe nodepair

nsincethe sameamountofsourcecurrent

ersnodeb,whilenonet sourcecurrententers

ndh.

ertingacurrentsourcein serieswitha

tuponthenetworkgeometryasdoes theopen-

fthatbranch.In thisalterednetworkthe

rossthenodepairoriginallylinkedby the

ormof severalidenticalsourcesbridged

anchesjoiningthis nodepair.

wemayregarda currentsourceasagen-

nopencircuitweunderstanda branchfor

dependentofthe branchvoltage;andbya

andabranchforwhichthecurrentis t,inde-

age.Fori,= 0,thecurrentsourceis identical

ter mayberegardedasadead currentsource.


/ h t t p : / / w w w . h




at,so longasvoltagesourcesappearonly

dcurrentsourcesareassociatedonlyin parallel

depairs,theirpresencedoesnot disturbthe

ensethatall matterspertainingtothat

ed,suchasthenumbersofindependentvoltages

racterizingthestateofthenetwork,ortheir

ranchcurrentsandvoltages.Ina sense,the

acurrentsourceandtheshort-circuitcharacter

eevident

soningof^*

gpara-ab

withFig.9. Passivebranchwithassociated

ndcan

placean opencircuitiftheinsertedsource

rcuitiftheinsertedsourceis avoltage.

eometryiscarriedout,thesourceappears

llelwithabranch(orwith severalbranches)

thabranch(or withseveralbranches).

mentsalone,therefore,areallthatneedto

ingdiscussion.

anchin anetworktohavethestructure

nka-brepresentsthepassivebranchwith-

andcurrentsources;thatis tosay,whenthe

suallyare formostofthebranchesin anet-

ucestothislink a-balone.However,we

hispointthatanyor allofthebranchesin a

vethe associatedsourcesshowninFig.9.

edas ageometricalconfigurationofactive

es.Thisturnofeventschangesnothingwith

saidpreviouslyexcepttherelationsbetween

chcurrents[designatedastherelations(b)

egardingtheformulationofequilibrium

netvoltagedropandthenet currentin

pandcurrentin thepassivelinka-b(noting

.9)are (vk+e,k)and(jk+ i,k)respectively.

that

ffected

eisplaced

chin

e.


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



TONS

atare relatedbythepassivecircuitelement

nts.Ifthefunctionalrelationshipbetween

the passivelinkisformallydenotedby

e,for thegeneralactivebranchofFig.9,

* +*',*)=y (t>* +e,*)(43)

notationz(j) reducessimplytoamulti-

thebranchresistance,andy(v)denotesa

gedropvby thebranchconductance.In

anchesthesymbolsz(j)andy(v)alsoinvolve

gration,aswillbediscussedin detaillater

theseelementsareconsidered.Forthe

ualizethe significanceofEqs.43withregard

ne.

referencetothe arrangementinFig.9,

obtainedifthecurrentsourcei,kis assumedto

sivelinka-baloneratherthanwith theseries

dthevoltagesourcee,*.I fi,k=0,the link

ltagesourcealone;ife,k=0,one hasthe

ebranchactivatedbyacurrentsourcealone.

gementreducestotheusualpassivebranch.

tions43aresufficientlygeneraltotake care

ncebetweennetbranchvoltagesandcurrents

entdiscussions.

eeffectofsourcesin thederivationof

oweasilystated.Namely,oneproceeds

epreviousarticlesfor theunactivatednet-

onsbetweenbranchvoltagesandbranch

theformof Eqs.43,soasto takeaccountof

eorcurrentsources.Thisstatementapplies

ofequilibriumequationsonthe looporthe

ssofthenatureand distributionofsources

heprocedureremainsstraightforwardandis

the unexcitednetwork.

resforDerivingEquilibrium

tobringtogetherincompactsymbolic

etting upequilibriumequations.Thuswe

awequationsintermsofbranchvoltages:


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



EDURES

branchvoltagesandbranchcurrents

ermsof theloopcurrents:

dule(like13,forexample)placeinevidence

ethecolumnsof thisscheduleyieldthebranch

opcurrents,Eqs.46.Theexpressionsforthe

s.45,areobtainedfroma knowledgeofthe

eassociatedvoltageandcurrentsources,as

uationsaretheKirchhoffEqs.44 expressed

nts.Oneaccomplishesthisendthroughsub-

Eqs.46intoEqs.45, andtheresulting

.44.Noting thatthelinearityofthe network

+i,k)=z(jk)+ z(i,k),theresultofthis

4,45,46leads to

k)]=etl(47)

dablelookingresultisaided bypointing

thepassivevoltagedrop inanybranchk

oop currentsirinthat branch,andthatthe

thealgebraicsummationofsuchpassive

ndatypicalclosedloopI. Theright-handside,

esymbole,i,is thenetapparentsourcevoltage

isgivenbyan algebraicsummationofthe

thebranchescomprisingthisclosedcontour

lvoltagesinducedinthesebranchesbycurrent

eouslybeassociatedwiththem.Thelatter

sentedbytheterm— z(i,k),mustdependupon

ionsinthesamewayas dothepassivevoltage

urrents,exceptthattheir algebraicsignsare

erises.

umEqs.47 statethelogicalfactthatthe

nanyclosedcontourmust equalthenetactive

r.If weimaginethattheloopsare deter-

treeandidentifyingthe linkcurrentswith

interpretthesourcevoltagese,i asequivalent

at, ifactualvoltagesourceshavingthese

ks andalloriginalcurrentand voltagesources

loopcurrentsremainthesame.Or wecan


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



TONS

thevoltagese,iareplacedin thelinks,then

esbecomesneutralized,andthe resulting

hatis,the loopcurrentsorlinkcurrentsare

ldbe ifalllinkswereopened.

nterpretationofthee,i inthattheymaybe

ofthevoltagesappearingacrossgaps formed

manysituationstowhichthesimplified

t.6is relevant,thisphysicalinterpretation

tiese,isufficesfor theirdeterminationby

work.

cedureandcorrespondingprocessofphysical

ederivationofequilibriumequationsonthe

awequationsintermsofbranchcurrents:

hebranchcurrentsandbranchvoltages

ermsof thenode-pairvoltages:

ule(like20, forexample)placeinevidence

ethecolumnsof thisscheduleyieldthe

fthenode-pairvoltages,Eqs.50. Theex-

rmsofthevk's,Eqs.49, areobtainedfroma

arametersandtheassociatedvoltageand

tedinFig.9.

uationsaretheKirchhoffEqs.48 expressed

voltages.Oneobtainsthisendbysubstituting

ntoEqs.49,andthe resultingexpressionsfor

tthelinearityofthe networkpermitsone

k)+y(e,k),theresultofthis substitution

s to

k)]=i.n(51)

dablelookingresultisaided through

epresentsthepassivecurrentinanybranchk

node-pairvoltageseractinguponit,and

Eq.51 isthesummationofsuchbranchcur-

ypicalcutset;forexample,theset ofbranches


/ h t t p : / / w w w . h




en ifthenode-pairvoltagesarechosenasa

1, whichisabbreviatedbythesymboli,n,

currentfor thiscutset,forexample,iti sthe

ntenteringnodenin anode-to-datumsitua-

tis givenbyanalgebraicsummationofthe

dwiththebranchescomprisingthepertinent

urrentsinducedin thesebranchesbyvoltage

eouslybeactinginthem.Thelatter currents,

heterm— y(e,k),mustdependuponthecir-

nthesamewayasdothe passivecurrents

oltagesexceptthattheiralgebraicsignsare

presentaflowofchargeintothe cutsetrather

umEqs.51 statethelogicalfactthatthe

ranchesof acutsetmustequal thetotalsource

.If weimaginethatthecutsets havebeen

tingatreeandidentifyingthetree-branch

oltages,thenwecaninterpretthesourcecur-

cesbridgedacrossthe treebranchesinthe

tsourceshavingthesevaluesare placedin

hesandall originalcurrentandvoltage

resultingnode-pairvoltagesremainthesame.

egativesofthecurrentsi,nare placedacross

eeffectofall othersourcesbecomesneutralized,

esponseiszero; thatis,thenode-pairvoltages

ezero,thesameas theywouldbeifall tree

uited.

nterpretationofthei,n inthattheymaybe

ofthecurrents appearinginshortcircuits

ranches.In anode-to-datumchoiceofnode

rdedasthenegativesofthecurrentsappearing

cedacrossthesenodepairs,and anode-to-

eshavingthesevaluescanbe usedinplace

currentsourcesin computingthedesired

ysituationstowhichthesimplifiedprocedure

vant,thisphysicalinterpretationofthenet

fficesfortheirdeterminationbyinspection

orsettingupequilibriumequationswillnow

ecificexamples:Considerfirsttheresistance


/ h t t p : / / w w w . h




TONS

mentvaluesin part(a)arein ohms,andthe

amperes,e,=5volts(bothconstant).In

ork(elementvaluesinohms)andits graphshowingthe

.

eisshownthegraph withitsbranchnumbering

efine loopcurrents.

spondingtothischoiceisgivenin 52.The

wequations:


/ h t t p : / / w w w . h




branchcurrentsin termsoftheloopcurrents,

tivelytoEqs.44and46in theabovesummary.

atingbranchvoltagestobranchcurrents,

ciatethecurrentsourcewith branch5(we

eitwithbranch2),thenall branchesexcept

ospecialcommentisneededforthem.The

1 isvi=— e,-f-ji,andthe netcurrentin

ch5 isj6= i,+(v5/2),theterm(»5/2)

ohmresistancewhichisthepassivepartof this

valuesgivenabove,therelationsexpressing

ntermsofnetbranchcurrentsread

)

activebranchesareseento containterms

rrent.

uationsarefoundthroughsubstitutionof

sultingexpressionsforthev's intothevoltage-

operarrangementthisgives

rthe loopcurrents.Onefinds

/3, i4=5(57)

qs.54yieldsall thebranchcurrents

3, u=0, is=25/3, j6=5/3(58)

urrentinbranch5. Thatinthepassivepart

anj5bythe valueofthesourcecurrent,and

.


/ h t t p : / / w w w . h




TONS

orkgiveninFig.10by thenodemethod,

agesthepotentialsofnodesa andbrespec-

easa reference.Theappropriatecut-set

veus thecurrent-lawequations,

branchvoltagesin termsofthenode-pair

tivelytoEqs.48and50in theabovesummary.

thebranchcurrentsto thebranchvoltages,

=vi+e, andjs=i, +0.5»5,sothatthe com-

sreads

seofEqs. 55.

uationsarefoundthroughsubstitutionof

sultingexpressionsforthej's intothecurrent-

operarrangementonefinds

dto be


/ h t t p : / / w w w . h




ethencomputedfromEqs.61 tobe

10/3,v4= 0,v5=-10/3,v8=10/3(65)

mustberememberedthatthevalue ofviis

ngthevoltagesource.Thedrop inthe

5volts.

hallconsiderthenetworkgraph shownin

erieswith thebranchesarevoltageshaving

ncenetwork(a)withbranchconductancevaluesgivenby

irvoltagevariablesisindicatedin (b).

eforthisgraphb =10,n= 3,andI= 7,it

oosethenodemethod.Ageometricalspecifi-

esisshowninpart (b)ofthesamefigure.In

ningtothischoiceofnode pairsalastcolumn


/ h t t p : / / w w w . h




UATIONS

ng"picked-up"nodesisaddedtofacilitate

ction.

sscheduleone obtainstheKirchhoff

fa- fa=0

(67)

efollowingrelationsforthe branchvoltages

voltages:

ei+e3

e3(68)

nsideredtoberesistive.Letus assumefor

lowingvaluesinmhos:

=3,g5=4,

g=2,gi0=6

hebranchcurrentsintermsof thenetbranch-

dilyfoundby notingtheappropriateexpression

partofeachbranchand multiplyingthisby

ctance.Forexample,thevoltagedropin the

»i+10,in branch3itis t'3+2,in branch5

hus weseethat

Eqs. 68intoEqs.70 andtheresulting

qs. 67givesthedesiredequilibriumequa-


/ h t t p : / / w w w . h




ementtheseread

)

=3.93(72)

-voltagedropsmayreadilybecomputedusing

rrentsare thenfoundfromEqs.70.

enceofKirchhoffvoltage-lawequations,itmightbe

erofequationsequals I=b — n,andifcollectivelythey

tages,thenthey mustformanindependentset.Show

by constructingacounterexample.Thus,withregard

ph,considerequationswrittenforthecombinedcontoursof

3and4, 4and1.Althoughall branchvoltagesareinvolved,

donotform anindependentset.

tement:"Thenumberof independentKirchhoff

ualsthesmallestnumberofclosedpathsthattraverseall

ph shown,determinewhetherasetofvoltage-law

ollowingcombinedmeshcontoursisanindependentone:

+8+9), (1+4+7), (2+5+8),

(2+3+5+6), (5+6+8+9)

b.1, andsolveit.

tage-lawequationswrittenforthefollowingcombined

ndependentset(1 +2+3), (2+3 +4),(3+ 4+1),

dent:(1- 2),(2-3), (3-4),(4 -1)?


/ h t t p : / / w w w . h




TONS

network,arevoltage-lawequationswrittenforthe fol-

sindependent:

),(5 +1)?

4-5), (5-1)?

dual.Makeappropriatesketchesandanswerthe

ed.

equationswrittenforthemeshcontoursin amappable

dependentsetbyconstructingthedualsituationand

dingproof.Inwhichsituationisthe proofmorereadily

ob.1, Ch.1,andchoosebranches5,6,7, 8asconsti-

hes,whichbecometheclosedpathsuponwhichthelink

rchhoffvoltage-law'equations,andusetheseto express

f thetree-branchvoltages.Nowwriteavoltage-law

closedpath,say,forthe meshcombination(1+2— 3)

uationsubstitutetheexpressionsforthe linkvoltages

thatitreducesto thetrivialidentity0=0.

e situationdescribedinProb.9,andthus givean

ngthatnomorethann Kirchhoffcurrent-lawequations

eseries sourceisavoltage,andthe paralleloneisa

sareinvoltsandamperes.Thepassiveelementis are-

ated.

nciplewhichallowsustoadd separateeffects,treating

did notexist,andrememberingthatanonexistentcurrent

tratethecorrectnessofeachofthe followingrelations

43.Thusshowthat thegivenactivebranchisreplace-

wingones:


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



tout intheprecedingproblem,reducethefollowingto

ssiveelementwitha seriesvoltagesource,(b)anequiva-

twithaparallel currentsource.

Prob.12to thefollowing:

Prob.12to thearrangementofsourcesandpassive


/ h t t p : / / w w w . h




TONS

he centralsourceisacurrent.Theothersources

gesor currents,accordingtotheirseriesorparallelassoci-

ssiveelement.Elementvaluesarein ohms.Through

,reducethisproblemtooneinvolvingasingleloop current,

obtainthe fourcurrentsii,t2,is, uintermsof thisone.

he branchnumbersmayberegardedasalsoindicating

esinmhos.Constructtwocut-setschedules,oneforthe

es,ei=«i,e2 =t%«s= »s,andtheotheronefor the

d.

the definitionofvariablesandthesecondonefor the

hoffcurrent-lawequations,obtaintheequilibriumequa-

etricalparametermatrix),andsolve.Alternatelyobtain

equationsthroughuseofthefirstschedulealone.Solve

oussolutions.

edualtoProb.16and solve.

hmbranchesasforminga tree.

esourcesinthelinks alone.Setuploopequations,and

f currentsourcesacrossthetreebranchesalone.Set

ve.Obtainallcurrentsand voltagesinthepassive

andcheck.

esourcesin(a),first,by replacingthe—4-voltand2-volt

esbyrespectivelyequalsourcesin thelinksandcombining

oltagesourcesandconvertedcurrentsources;second,by

otingthenetvoltagesacrossthegaps thusformed(the

esarethenegativesofthese).Checktheresultsfoundby

rlyinpart(b) findthedesiredequivalentcurrentsources,

fvoltagetocurrentsourcesandthenreplacingcurrent

ualonesacrosstree branchesandcombiningthesewith

ebranches;second,byshort-circuitingallthetreebranches

sin theseshortcircuits(thedesiredcurrentsourcesare

ainchecktheresultsfoundby thetwomethods.


/ h t t p : / / w w w . h




rcesin(a) yieldthecorrectloopcurrentsbutthat the

anches,whicharenowpurelypassive,arenot theactual

Hence,ifweconvertthevoltagesourcesin(a)to equiva-

ansfertheseacrossthe treebranches,weshouldnotexpect

esfoundinpart (b).Similarly,wecannotexpectfrom

oseof(a) throughsourcetransformationmethodsalone.

roblem.

wsthegraphof anetworkconsistingofseven1-ohm

ce.Findthevaluesof thenodepotentialsei,ej,ej with

monnodeat0.Althoughanyvalidmethodisacceptable,

ethetechniqueofsourcetransformationsinorderto avoid

ofalgebraicequations.

wnconsiderbranches1,3,and4as formingatree.

ththe loopcurrents,andwriteatie-setscheduleforthe

citlythethreesetsofequations:(1) Kirchhoff'svoltage-

opriatevolt-ampererelationsforthebranches,(3)the

fthe loopcurrents.Substitute(3)into(2)and then(2)

briumequationsonaloopbasis.

tofequationsdirectly,usingmeshcurrentsasvariables

rediscussedinArt.6.


/ h t t p : / / w w w . h




TONS

Prob.20considerthenode-pairvoltagesfroma,6,

ependentset.Writeacut-setscheduleforthem.Then

uations:(1)Kirchhoff'scurrent-lawequations,(2)the

elationsforthebranches,(3)thebranchvoltagedrops

voltages.Bysubstitutionof(3)into(2),and thenthese

umequationson anodebasis.

tofequationsdirectly,usingthesamenode-pairvolt-

mplifiedprocedurediscussedin Art.6.

nts1,2, 3,4,11,as variables,repeatparts(a)and(b)

kshownhere.Branches1through10are2-ohmresist-

mresistancein parallelwitha1-amperecurrentsource.

associatedsourcesasshowninFig.9becomesdegen-

ssiveresistanceassumeaninfinitevalue,then itsvoltage

rrentisconstrainedbytheassociatedcurrentsourceto

wayofdealingwiththissituation istorevisethe network

hecurrentsourceasshown inFig.8.Show,however,that

thissituationbytreatingthis branchinthenormalman-

thistypeofdegeneracycreatesnoproblemsinceterms

wequationsinvolvingthecurrentjk=—i,k simplybe-

daretransposedtotheright-handsides.On aloopbasis,

ctthetie-setschedulesothat itsfirstI — 1rowsdonot

entifyingloopcurrenttjwith theknownbranchcurrent

1oftheloop equationssufficientforthedetermination

tration,treatthe followingcircuitinthismanner.Let

resistancevaluesinohms.

associatedsourcesasshowninFig.9becomesdegen-

ssiveresistanceassumeazero value,thenitscurrent

ltageisconstrainedbytheassociatedvoltagesourceto


/ h t t p : / / w w w . h




ewayofdealingwiththis situationistorevisethe network

hevoltagesourceinthe mannershowninFig.7.Show,

ernatelymeetthissituationbytreatingthis branchinthe

oopbasis thistypeofdegeneracycreatesnoproblem

fvoltage-lawequationsinvolvingthevoltaget,*— — e,*

antitiesandaretransposedtotheright-handsides.Ona

canconstructthecut-setscheduleso thatitsfirstn — 1

nch,thus identifyingnode-pairvoltageenwiththeknown

ringthefirstn — 1ofthenode equationssufficientforthe

wns.Asanillustration,treatthefollowingcircuitin this

mbersequalconductancevaluesinmhos.

theaccompanyingsketch,assumethebranchnum-

esistancevaluesinohms,andleti, beoneampere.Choosing

ks,findaset oflink-voltagesourcesequivalenttothegiven

enegativesofthe voltagesappearingatgapscutsimul-

hthesereplacingthe currentsourcei,,writedownby

equationsonameshbasis usingthesimplifiedprocedure

ngthenetsourcevoltagesaroundmeshesasthe right-

lyobtainthesesameequationsusingtheprocedurede-

hthecurrentsourceistreatedas anormalbranch,and

sourcesinparallel withbranches3and4;convertto

iththesebranches,andagainwrite meshequations.Will

currentsas above?Explainindetail.


/ h t t p : / / w w w . h




Methods

iumequationsforagiven network,the

ghtheirsolution.Hereonemayproceedin

hoicedependinglargelyupontheobjective

ne.Thus, onemaybeinterestedmerelyin

specificsituation,orin amoregeneralsolu-

fthenetworkparametersentersymbolically.

s actuallyequivalenttothesimultaneous

cificnumericalsituationsandconsequently

cdifficultieswhichcanbeovercomeonlythrough

ymoregeneralmethodsofanalysis.Aneffec-

uchproblemsisgiveninthe nextarticle.For

ernourselveswiththeless difficulttaskof

alcase.

exampletheEqs.24 appropriatetothenet-

harbitrarynonzeroright-handmembers,thus:

e3+0.500e4= 1

3e3-0.750e4= 2

e3+0.950e4= 3

e3+1.061e4= 4

odofsolvingasetof simultaneousequations

maticallyeliminatingvariablesuntilanequa-

nisobtained.Afteritsvalueisfound,an

doneothervariableis usedtocomputethe

n,andsoforth.Unlessthe entireprocessis

considerableamountoflostmotionmay result.

aneffectiveone.


/ h t t p : / / w w w . h




METHODS

mericalcoefficientsenterintothecomputa-

reitissensibletoomit writingthesymbols

dconsideronlynumericalmatrix2.Wenow

1.000

02.000

3.000

14.000

mannerinwhichonemaycarryoutthe follow-

s.1:First, weundertaketoeliminateeifrom

uations;thisstepleavesus withthreeequa-

romall butthefirstof these,wenoweliminate

quationswithe3ande4.Fromone ofthesewe

gle equationine4.

alsohaveanequationinvolvinge3and e4,

4,and thefirstofthe originalequationsin-

Wecan,therefore,readilysolvetheseequa-

inall theunknownswithoutfurtherdifficulty.

theequationine4alone.Next,the onein-

dfore3.Then,withe3and e4known,theequa-

4 yieldsthevalueofe2,and thefirstofthe

susedtofind ei.

theprocessof eliminatingeifromall

dentlyequivalentto aneliminationofthe

ementsinthefirst column.Thisendis

gdirectlyupontherowsofmatrix2as one

dingEqs.1.Thus,ifwe addtotheelements

ectiveas-multipliedelementsofthefirst row,

esultreads

2.855

secondrow.Similarly,anewthirdrowis

ementsof thepresentthirdrowthe respective

hefirstrowwitha =—0.643/1.142,yielding

2.437

analogouslyformedwitha=—0.500/1.142,

3.562(5)


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_ u s e # p d - g

o o g l e



ANDRELATEDTOPICS

edbyobservingthatthe originalmatrix2

dintothefollowingequivalentone:

heequationscorrespondingtothismatrix,it

henumericaloperationsjustcarriedoutare

onofeifrom thelastthreeofthe original

atefromthelast twoequationscorre-

s endweaddtothe elementsofthethird

pectivea-multipliedelementsofthesecond

obtainingthenewthird row:

.008(7)

entsofthesecondrowin 6bya= 0.323/1.492

veelementsofthe fourthrowgives

.180(8)

sassumedtheform

quationsin whicheidoesnotappearin the

notappearinthethird andfourth.

quivalenttoeliminatinge3fromthelast

presentedbythematrix9byadding thea-multi-

rowtotherespectiveonesofthe fourthrow,

vingafinalfourth rowthatreads

.251(10)

mfor thematrix:


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i g i t i z e d

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_ u s e # p d - g

o o g l e



METHODS

eequation

mpliestheequation

8(14)

alue13 fore4,becomes

4.150(15)

(16)

trix11 wenexthavetheequation

e4=2.855(17)

3and16,

+0.732 =12.216(18)

orrespondingtothefirstrowin matrix11,

re2,e3, ande4alreadyfound,becomes

1.135=1(20)

4(21)

stematiceliminationmethodisthe trans-

atrix2i ntotheso-calledtriangularform11,

eobtainedthroughanobviousrecursionprocess

putationofthelastof theunknowns•••e4

theothers.Itmay readilybeseenthatthis

einvolvesaminimumoflost motionandhence

yinanynumericalexample.

equationsalreadyhavezerocoefficients,it

earrangetheequationsin orderthatthe

plicableinpreciselytheformdescribedabove

advantageofthe simplificationsimpliedby


G o o g l e - d

i g i t i z e d

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_ u s e # p d - g

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NDRELATEDTOPICS

modificationsinprocedure,however,the

orhimselfas hecarriesoutactualexamples,

ssionofthemwill notbegivenhere.

methodofsolvingsimultaneousalgebraic

umericalexamples,theamountof computa-

aterthanin thesystematiceliminationprocess

theotherhand,affordameansfor expressing

tsymbolicformthatenablesonetostudy their

husdeducewithlittle effortanumberof

ralnetworkcharacteristics,someofwhich

atterpartofthis chapter.Ourimmediate

y someofthemoreimportantalgebraic

s.

tofthesystemofequations

=yi

n=2/2

=yn

kethecorrespondingmatrix(differingonly

entsisenclosedbetweenverticallinesinstead

ts algebraicsignificanceitisentirely different

a functionofitselementsandhas avalue

softheseelementsas doesanyfunctionof

mentsarethecoefficientsa,kinEqs. 22.For

nthasn2elementsand issaidtobe ofordern.

ularkind offunctionofmanyvariables

maticiansforthesole purposeofitsbeing

multaneousequations.Henceitwasgiven

utto servebestthisobjective.Thesemay


/ h t t p : / / w w w . h




wingthreestatements:

tis unchangediftheelementsof

dedto therespectiveonesofan-

tis multipliedbykifall theele-

mnaremultipliedbyk.

tis unityiftheelementsonthe

yandall othersarezero.

writtenin theform

operties24and25it followsthatthevalue

thefc-multipliedelementsofanyrow (or

espectiveonesof anotherrow(orcolumn).

ynegative,thisstatementincludesthesub-

itionofrespectiveelements.Italsofollows

adeterminanthasthe valuezero(a)ifthe

umnare allzero,or(b) iftheelementsofany

espectivelyequalorproportional,fora row

k =0,anda conditionofequalorpropor-

mediatelyleadstoa row(orcolumn)of

manipulationsofthesortjust mentioned.

eterminantmayreadilybefound through

ce,bymeansof them,onecanconsecutively

iagonalelements(afterthefashionthat

rticleistransformedtoform11).Oncethe

onalform,properties25and26 showthatthe

fthediagonalelements.In factitcanbe

ntintriangularformhas thissameproperty;


G o o g l e - d

i g i t i z e d

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ANDRELATEDTOPICS

elowthe principaldiagonal(upperleftto

edto zerothroughappropriatemanipulation.

eevaluationofa determinantishadthrough

opmentorexpansion.Thusthevalueofa

asthesumof productsofthesuccessiveele-

mn)andquantitiesreferredtoascorresponding

ctionweshalldescribepresently).Denoting

anelementa,kby A,k(thatis,bytheupper-

s),wemayindicatetheLaplaceexpansionof

Ain(29)

a2nA2n(30)

s

\Ani(31)

erentpossibleexpansionsofthis sort,and

ericalvalue,namelythevalueofA.

rA,kconsistsin canceling(thatis,re-

kfrom thegivendeterminantAandpre-

daminordeterminantoforder n— 1)by

ingfactorJ JJ***.Thusthecofactorequals

rminantexceptforitsalgebraicsign,whichis

referringtothecanceledrowand columnis

mofindexesisodd.The canceledrowand

tersectsatthepositionofthe elementa,k

ssociatedineachtermof theexpansion.In

retheelementsofadeterminantaremerenum-

exesto identifytheirrowandcolumnposi-

bservedbyinspectionindeterminingthe

ustermsin theexpansion.Thesameistrue

icformwherethe notationforonereasonor

anevidentmanner,informationasto the

ofits elements.

pressesthevalueof agivendeterminantof

uesofn determinants,eachofordern— 1.

tscanseparatelybeexpressedinterms of

eterminants,eachofordern— 2.Con-

finallyyieldsthedesired resultwhenthe

nedareoforder1.Sincethe numberofterms


G o o g l e - d

i g i t i z e d

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_ u s e # p d - g

o o g l e



givenbyEqs. 29,30,or31)is n,andeachof

)furthertermsofwhicheachagainyields

,it isreadilyappreciatedthatthefinalevalua-

chof whichisaproductofn elements.

luationiscomputationallyverytedious.

ornumericalevaluationbutratherforthe

apropertyofdeterminantsthatenablesone

symbolicform)thesolutionstoagivenset of

Thefollowingdiscussionnowshowshowthis

.

hatthecofactorsassociatedwiththeele-

olumn)donotdependupontheelementsof

e simplereasonthatthisrow(or column)

nof thesecofactors.Sofarasthevalues

cerned,theelementsintherespectiverow(or

aluesatall.With thisthoughtinmind,

xpressionlike

ainA2n(32)

edevelopmentofthedeterminantA,except

st rowaremultipliedrespectivelybythe

theelementsofthesecondrow.Thatisto say,

ntsan,012, .••,ain aremultipliedbythe

ofthewrongrow.Sincethesecofactorsdo

entsofthe secondrow,theentireexpression

hese elements,andhenceonecansuppose

minanthavingitssecondrowidenticalwith

nant,however,itrepresentsthe correct

ngitssecondrow;and,sincethisdeterminant

enthatexpression32,evaluatedfor anydeter-

o.

thesumof productsformedfromthe

umn)ofa determinantandthecofactorsof

fanotherrow(orcolumn)equalszero.That

whatappearstobe theLaplacedevelopment

ciatetheelementsofa row(orcolumn)with

ftheelementsofanyotherrow(or column),

yszero.

supposethatwemultiply thesuccessive

ofactorsAn,A21,Ani(associatedwiththe

n),andthenadd allterms,acolumnata time.

dstoxi timesaLaplacedevelopmentofA;


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ANDRELATEDTOPICS

ox2times oneofthesepseudodevelopmentsof

sameappliestoallthe othersumsofleft-

tasa result

Aniyi(33)

sisobtainedin asimilarmanner.More

btainanexpressionfor theunknownXk,we

veEqs.22aremultipliedrespectivelybythe

sociatedwiththeelementsofthefcth

dditionofalltermsyields zerosforthe

x'sexcept£*,whichhasfor itscoefficient

opmentofAalongitsfcthcolumn.Hence

nforXkgiven below:*

sthatonlyoneofthe quantitiesyi••• ynis

workisexcitedatonlyonepoint).I fthe

orrespondingunknownXkbecomes

ear,theresult whenally'saresimultaneously

oftherespectiveresults36for separatenon-

terpretthemoregeneralresult35 asasuper-

they'stakenseparately.

oset downincompactanalyticformthe

e(voltageorcurrent)at anydesiredpointina

itationappliedat thesamepoint(s= k)

).Asalreadymentioned,noadvantageis

ericalevaluationisconcerned;infactitisusu-

ericalcasestouse thedeterminantmethod.

mofresult36,in conjunctionwithknown

nablesonetodeterminebyinspectionnumerous

orksolutionsandcharacteristicnetwork

t variouspointsinourlater discussions.

hisresultis knownas"Cramer'srule."


/ h t t p : / / w w w . h




ECIALNETWORKCONFIGURATIONS121

dderandOtherSpecialNetwork

workgeometryordistributionofparameter

eveshortcutsin numericalcomputation,itis

naflexibleattitudetowardthegeneralsystem-

rareoftentheonly onesapplicableinrandom

ereforewell tobethoroughlyfluentintheir

suchuse tobecomehabit-formingtothe

warenessofcomputationaleconomiespossible

htheuse oflessconventionalproceduresof

thisconnectionitiswellalso torecognize

ancedladdernetwork.

mnetworkismoreofa theoreticalfictionthan

workconfigurationsfoundinpracticeare

plicityofstructureorageometricsymmetry

s(evenas totheirmodeofexcitation)that

oajudiciouslychosenmethodofattack.

dnetworkconfigurationofthissort isthe

dermentionedinArt.9ofCh.1 andillustrated

gure1showsaspecificexampleofsucha

ebrancheswiththeohmicvaluesindicated.

sumedtohavezeroresistance)servesasa

rentsinthe variousshuntbranchesandin

formofacommongroundedreturnconductor

nofas being"unbalancedwithrespectto

efortheexcitationvoltagee,is given,andthe

agesinthevarious branchesarewanted.

stoignore thegivene,valueto startwith

,the potentialoftheright-handnode,is1 volt,

valueof e,givessucharesult.

at,ife5 =1,thenjio =jg=1 ampere,

4=e5+*■9=2volts.Nextjs=e4/2=1

=2amperes.Thefollowingrelations,


/ h t t p : / / w w w . h




ANDRELATEDTOPICS

nthusestablished,shouldbeself-evident:

attheinputto obtain1voltatthe output,

=0.0099.Ife,is givenas50volts,thenthe

0/101)volts,andallthecurrentsand voltages

rethevaluesfoundabove,multipliedby the

tisasimple mattersubsequentlytoadjust

modateagivenexcitationvoltagee„while

ionsitiseasier toassumeonevoltat the

tresistanceise,/ji=101/57 =1.77ohms,

ilyobtainablebythis method,whichisfar

odeanalysis.Whereverladdernetworks

waysfollowasimplifiedprocedureofthis

ventionalmethodsisindicatedforspecial

conditionsenableonetosolvea problem

eofthissortis thefollowing.Considera

ebranchesarearrangedgeometricallyasare

ssumeeachbranchtobearesistanceof1 ohm,

atoffindingthenet resistance(a)between

rsofanysideofthe cube,(b)betweendiagon-

ecubeitself.

pedupona planesurfaceasshowninFig.2.

volvesfindingthenetresistancebetweennodes

b)thenet resistancebetweennodesaandcis

currentisinjectedintonode aandretrieved

metryitisimmediatelyclearthatnodese,f, g,

mepotential,namelythearithmeticmeanof

d d.Wemayconclude,therefore,thatthe

7 arezero.Removalofthesebranchesleaves

ennodesaand d,viz.:a2-ohmpathvianode


/ h t t p : / / w w w . h





e,anda3-ohmpath vianodesbandc.The

to be

m,weassumethatacurrentis injectedinto

odec.Undertheseconditionsthesymmetry

conclusionthatthecurrentsin branches

gtothe

thegraph

mputationof

odesaandc.

osein branches8,9,10.Consequentlynodes

entialandmaybeallowedtocoincide;and,

eequal potentialsandmaybeallowedto

ngfromthesuperpositionofthesegroups

.Notingthateachbranchis a1-ohmresist-

nspectionthatthe desiredresistanceisgivenby

9)

etoseem somewhatmorecomplexthrough

branchesdiagonallyinsidethecube.With

procedureamountstoaddingabranchfrom

frometog, andonefrom/to h.Theprocess

wever,assimpleasbefore.Thatis tosay,

eagaintoconcludethatnodesb, e,hhavethe

sethatnodesd,f,g havethesamepotential,

againapplies,exceptthatwemustinsert

ches.Oneoftheseconnectsawithc;the


/ h t t p : / / w w w . h




ANDRELATEDTOPICS

withthecentralgroupofbranches.Hence

pletheresistancenetworkshowninFig.4(a),

hateach branchhasaresistanceof1ohm.

osenas adatum,andletthe potentialsei••• es

ork(a)anditsdegenerateform(b)whenexcitedby the

threspecttothis datumbeunknowns.The

onsistoftwocurrentsourcest,i andi,2as

returnthroughthedatumnode);theirnumeri-

mentofnointerest.

expeditiouslycomputedbyaddingthe

akenseparately.Consideringthesourcei,i

thesymmetrythatnocurrentexistsin the

ge3ande4,andsowe mayconsiderthisbranch

etworkconsistsoftwoidenticalhalves,

calcenterline.Onesuchhalfis shownsep-

Notethatthe resistorbetweeneiandthe

sbeingsplit longitudinallydownthemiddle,

osssectionand doublingitsohmicvalue;the

alved.

nowrecognizedtobeanunbalancedladder.

stributionalongit,weassumethe output


/ h t t p : / / w w w . h





yinspectionweseethate5is 2volts,and,since

etweenei ande5is2 +1=3 amperes,

currentfromeitothe datumisej/2=5/2

+(5/2)=11/2amperes.Thuswe see

sei-■5,cs — 2,e4— 1;or*,j= 1ampere

,e4=1/11volt. Hence,returningour

orkinFig. 4(a),wecansaythat

, e3=e4=1/11volt(41)

oni,2alone areentirelysimilar,withashift

Onecansayatoncefromthevaluesgivenin

1, ei=e5=1/11volt(42)

at actuallyi,i=5amperes,andi,2 =10

rethe onesin41multipliedby 5plustheones

3=55/11,e4=25/11,

byvoltagesourcesasshowninFig.5, the

cablesince,

h.2,wecan

valentsetof

s.Thuswe

placethe

entcurrent

airs(in this

whilethe

resimul-

nodepo-

definitionof

current

enode-

othing is

shortcir-

odepairs; and,ifwenowremovethecur-

e networkstillcannotchangebecausethe

cited


/ h t t p : / / w w w . h




ANDRELATEDTOPICS

pairvoltagesat thezerovalue.Theshort

efunctionspreviouslyperformedbythe

ycarrythe currentsthatthesesourceswere

.* Henceonecanfindthevaluesof the

sbycomputingthenegativesofthecurrents

rcuits(in thenode-to-datumdirection)with

applied.

pertinentshort-circuitcurrentsareseen

— e,2,t',,4=0,i„5 — 0(44)

saregiven ase,i=—10volts ande,2=15

esamenodepotentialsei .••e5 resultif

dtonodes1, 2,and3havingthe values

5 amperes(45)

solution involvingthesituationinFig.4

icable.

thattheresultingvaluesforthe e'sthus

neswithreferencetothe givensituationshown

nthisfigureis thepotentialofthetopnode

andthe voltageacrossthe1-ohmresistor

— e,i.Similarlythevoltageacrossthe1-ohm

ectinge2ande3 ise2— e3— e,2,andsoforth.

heseexamplesillustrateisthe simplifica-

out intheanalysisofcertainnetworks

mewayanybranchesinwhichthecurrent is

rivialbranchesoftenrevealsthe remainder

n.In ananalogouswayonecancapitalize

tingfromthediscoverythattwonodesinthe

als,fortheiridentification(connectionby

hichis obviouslypermitted,canlikewise

utsinnumericalcomputation.

spointhow itisthat ashortcircuitcandothe work

iscapableofsupplyingenergyandthe shortcircuitis

thesituationdescribedhere,thecurrentsourcesare

andhencearenot calledupontodeliverenergy.Inthis

aswell.

ayalternatelybe obtainedthroughconversionofvolt-

esistancesinto currentsourcesinparallelwiththem(as

Art.5below)andsubsequentlyreplacingtheresulting

nd2by onefrom3to datumandanotherfromdatumto

heckthe resultsgiveninEqs.44.


/ h t t p : / / w w w . h




ALENTS

rksagain,itis interestingtoseethatthe

ussedabovelendsitselfalsotocertainkindsof

oseonewishestoobtainastatedvoltage

networkasindicatedin Fig.6wherethe

ethedesirednodepotentials(herechosenas

tegers).Thebranchresistancesr0• ••r8are

excitationtobe attheleft.

ownsthanspecifiedquantities,itis clear

esistancesarbitrarilyandthusarrive atmany

eepthingsverysimple,supposewechoose

equalto1, 4,9,16,25ohms.This choice

withdesiredvaluesofnodepotentialsspecified.

shunt branchesequalto1ampere,whencethe

ors ritr3,r5,r7 areimmediatelyseentohave

resrespectively.Intermsofthesebranch

tentialdifferencesinthesebrancheswehave

gresistances,namely,rt=3,r3 =5/2,

Theinputresistanceisseento be5ohms.

hisnetworktotheleft, uptoanode potential

storsincreaseupton2ohms,the input

ms,andtheseriesresistorssettle downtoa

ns;Wye-Delta(Y-A)Equivalents

nationmethod(discussedinArt.1)is

efulto interpretthesuccessivestepsinterms

ationsintheassociatednetworkgeometry.

esentequilibriumonthe meshbasis,the

meshcurrent)may,underappropriatecircum-

ometricallyasequivalenttotheeliminationof

omthenetworkgraph;and,ona node-to-

onofanode-potentialvariablecorresponds

associatednode.Anunderstandingofthe

relationbetweencertainanalyticmanipula-

limplicationsisoftenfoundusefulboth asan


/ h t t p : / / w w w . h




ONANDRELATEDTOPICS

onandin thephysicalinterpretationofnet-

cussionofthese ideaswouldseem,therefore,

rkfragmentinvolvingthreebranchesand

nthebranchesindicate thecorresponding

os,andthepotentialsei •••e4 arespecified

mondatum,notshown.

ediscussedinArt. 6,Ch.2,oneobtains

kingthefourvoltagevariablesei ••.e4,dis-

therest ofthenetworkinwhichthis fragment

0

0

0

rix

natethevariablee4 fromtheseequations.

ulationsofthematrix[G]aresuch aswill

eeelementsinthefourthcolumn.This result

tothe elementsofthefirstrowthe (2/10)-

fourthrow,thenaddingto theelementsofthe

tipliedelementsofthefourthrow,and finally

he thirdrowthe(5/10)-multipliedelementsof

gconductancematrixbecomes


/ h t t p : / / w w w . h




ALENTS

the lastrowfromitselfand reduceallits

tep istrivialandunnecessarysincethisrow

nceanyway.Thefirstthreerowsdetermine

revisednetworkinwhichnode4 isnolonger

3.Wemaysaythatthroughthe above

beendecoupledfromtherestof thenetwork,

enaccomplishedinsucha waythatthere-

reunaffected.Thenetworkfragmenttowhich

onsistingofthefirstthreerowsand columnsof

nasawyeFia.8.Configurationknownasa detta

ernetwork,circuit,equivalenttothewyein Fig.7.

s.Elementvaluesarein mhos.

erthatmatrix47pertainstothe networkportion

beregardedasequivalenttothatin Fig.7,

eplaceonebytheotherwithoutdisturbing the

orkinwhichthis fragmentisembedded.

eometryisreadilyseentobe thatshownin

calvaluesagainareconductancesinmhos.

ultis readilycheckedbywritingdown,forthe

spondingnode-conductancematrix,according

ereferredtoabove,andcomparingit withthe

nsofmatrix 48.

manipulativeprocedurethattransforms

ngthecorrelationofthelatterwith thenet-

mulatea directprocedureforthetransforma-

7intotheequivalentformshownin Fig.8.

llytheelementsof[G]byg,kand thoseof

hemanipulationsleadingtotheelementsof


/ h t t p : / / w w w . h




ANDRELATEDTOPICS

pressedby

totheremainingg',kare evident,andso

lof thesedown,especiallysinceweobserve

eresultingnetwork,Fig.8,dependsonly

oefficientsg'i2<0'13,0'23>thenegativesof these

ftherespectivebrancheslinkingthenodes

er.Notingthatthe correspondingcoefficients

zero,* wehaveforthebranchconductancesin

ofsummarizingthesesimpleresultsisshown

etworkfragmentofFig.7is drawnwithfull

inFig.8)is indicatedbydottedlines.The

ubscriptsareconductancesinthegivennet-

r-caseliteralsubscriptsrefertothe branches

Applicationoftheresults50, notingthat

enetworkin whichthepartconsideredhereliesem-

mayofcoursehavenonzerovalues.Inanycase,their values,

he remainingnetworkstructurethattheyimply,remain

redthroughoutthesemanipulationsandhencemayforthe


/ h t t p : / / w w w . h




ALENTS

hegivennetworkasa Y(wye)andtoits

saA(delta),forreasonsthat areself-evident.

tionoftheFig.10.The star,whichisagenerali-

bedbysayingthatany conductanceinthe

eproductofthetwo adjacentwyeconductances

hreewyeconductances.Toapplythisso-

mation,therefore,oneneedmerelyfollowthe

descriptivephrase:"theproductofthead-

sumof allthree."

owever,ofthe importantfactthatthis

oninthe networkgeometryaccompanying

liminationprocedurewherebyoneofthe

)issuppressed.Thustheeliminationofa

nttothe eliminationofanode.Througha

esameprocedureonecanultimatelyrender

implethatthedesired responsemaybeseen

pleofthistypeof transformationinvolves

gentfromthenodein question,thesame

es,regardlessofhowmanybranchesarein-

nageneralsituationinwhichn — 1branches

expressedby


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ANDRELATEDTOPICS

noden thatistobe eliminated.Interms

,.••, gn—iofthesebranchesandthenode

2)

the matrix[G]correspondingtothissituation

enis accomplishedbyaddingtotheele-

espective(17,/pnn)-multipliedelementsofthe

aluess=1,2, •••,n — 1.Inthenetwork

ngconductancematrix[(?'],itis important

theelementsinanyrowof [G]iszero,and that

yedbytheindicatedrowtransformations.

elementequalsthe sumoftheabsolute

elementsinthesamer ow;andintheassociated

timpliesthatthereareno brancheslinking

datumnode (somenodeotherthananyof

eofthegivenconfigurationinFig. 10.That

node-conductanceparametersg,kin[G]and

os ^kneedbeconsideredin theprocessof

heitherofthesematrices.Allbranchesare

onductancevaluesequalto—g,kor—g',k

subscriptsrefer.

ulationstowhichmatrix53is subjected,

quivalenttothat showninFig.10)in which

eachoftheremainingnodestowhichthe

—irefer,islinkedwith everyothernodeinthis

conductancegivenbyformula54.The

spokenofasastar, andtheresultingone

o astheequivalentmesh.Thetotalelimina-

e,calledastar-meshtransformation,andis

ofthewye-deltatransformationdiscussed


/ h t t p : / / w w w . h




ALENTS

eraltransformationincreasesthetotalnum-

thenetwork,it ismoreoftheoreticalinterest

is

appli-

carry-

attern

ch

ona

elimi-

mthe

acor-

net-

milar

nse

of

ed.

work

sona

depictedin Fig.7forthenode basis.

dina

relatingthefourmeshcurrentsinvolved,

mentsofthe sthrowtherespective(r,/r44)-

astrow,for s=1,2,3,one obtainsamatrix

eelementsofthefourthcolumnarezeros.

histransformedmatrixtherecorrespondthree

hemeshcurrents*'i,i2,and 13;thatisto say,

kethoseof [R]havethepropertythatthe

wequalszero,or thatthediagonalelement

olutevaluesofthenondiagonalelements.

pondingto[R']isfoundby consideringonly


/ h t t p : / / w w w . h




ONANDRELATEDTOPICS

7)

sofbranchescommontomeshessand k.

ComparisonwithFig.11 suggeststhatthe

edesignatedasadelta-wyetransformation;and

evaluesin theresultingwyeindicatesthat

toftheFig. 13.Pictorialrepresentationofthe

luesare delta-wyetransformationexpressedby

qs.58.

denticalwiththatcharacterizingthewye-

onlydifferencebeingthatresistancevalues

aluesareinvolved.

presentresultafter thefashionthatFig.9

tfeaturesofthewye-deltatransformation.

ticrelationsare

phrase "theproductoftheadjacenttwo,

hree"isagainapplicable.

ceduresoas toinvolvetheeliminationofa

bymorethanthree othermeshes(thelogical


/ h t t p : / / w w w . h




ALENTS

howninFig.11), althoughstraightforward

mitageometricalinterpretationforthereason

nstructagraphhavingan arbitrarynumber

eshindependentlyhasabranchin common

thofthisstatementmayalsobe seenfrom

esultingfromthe eliminationofthecentral

fFig.10is notmappableonaplaneand hence

oninArt.9 ofCh.1)doesnot possessadual.

orksoughtinthegeneralizationofthe mesh-

nonexistencepredictsthefutilityofany

neralization.

dertoappreciateabitmorespecificallythe

etusconsideran examplerepresentingthe

minationprocedurefromthesituationinFig.11

shesto oneinvolvingfive.Throughchoosing

gurationfortherestofthe networkinwhich

erationisembedded,wefindthattheelimina-

sageometricalinterpretation,althoughnot

noptimisticattitudemightleadusto expect.

andnot verypractical,butdoesservein

etheprinciplesinvolved.

g. 14wherewehaveincludedalsothe

transformationwhichisits dual.Sincethe

ertounderstand,itsinclusionin thetotal

tailsofthe mesh-starsituation.Ineach

chthetransformedportionliesembeddedis

eshtransformationeliminatesnode5,while

tioneliminatesmesh5.Asthegivennetwork

s drawn,mesh5istheperipheryand hence

e simplecharacterthatameshshouldhave,

worktobe drawnonasphere(as explained

readerisencouragedtodoanywaysincehe

asilythemappablecharacterofall thegraphs

tureofthemesh-startransformationisthe

the restofthenetworkdoesnot remain

shtransformation.Howeverwedoseethat

oneless intheresulting"star"than inthe

egeometrycorrespondsineverywaytothe

edintheeliminationofmeshcurrent5.The

1,2,3,4) traversethesamefixedbranches

tionsafterthe transformationiscarriedout


G o o g l e - d

i g i t i z e d

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_ u s e # p d - g

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ANDRELATEDTOPICS

ancesinthestar aregivenbythelogical

ndthedualityof thesenetworkswithrespect

nthe star-meshexampleisborneoutinthis

entformulasare identicalexceptforan

andconductance.

at thispointwhatwegain byallthis

nofthesystematiceliminationproceduredis-

ustcarryoutthisprocedurenumerically

thestepsgeometrically,especiallysincesuch

omecasesinto exceedinglycomplexandeven

eansweristhatwecan gaincomputational

oususeoftheseideasinappropriatesituations.

encetoknowhowandwhento usethem.Per-

servebest toillustratetheseremarks.


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



sinthe solutionofaproblemthroughrepeateduseof the

.Thedarkenednodeineachsketchis theonethatis elimi-

inmhos.

ussay,isgiven,and thevoltageratio

writing theequilibriumequationsandsys-

riables,onecansavetimeandwritingeffort


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ANDRELATEDTOPICS

sdirectly.Thus,throughuseofthewye-delta

minatetheblackenednodeinnetwork(a)and

htheelementvaluesare inmhos.Nextone

waythe blackenednodeinnetwork(b)and

nto obtain(d)from(c)and finally(e)from

ormationsonehaskeptto aminimumthe

putation,foronlythoseportionsofthe total

taresuccessivelyinvolvedinthe specific

vingatthenetworkshowninpart(e) of

an readilybeobtainedthroughapplyingthe

eviousarticle.

wouldnotbeas usefulifultimatelyonealso

entialsofsomeof theeliminatednodes,

procedureforaccomplishingthisendsubse-

asto howandwhentoapplythedirect

eswilldependonthedetailedaspectsofeach

dbe evidentfromtheaboveexamplethatthe

esituations,.saveanappreciableamountof

Theorems

cproceduresfortakingaccountofthe effect

isshown(byEqs.47 and51)thatthe actual

ontheloopbasis byanequivalentsetof voltages

ops)ofthe network,andonanodebasisby

tsi,nbridged acrossthetreebranches(or

Usingeither setofsources(e,iori,n ac-

por nodemethodofanalysisischosen),the

thatonewillobtainthe samevoltagesand

etworkasthoseresultingfromthe actual

aybeamixtureofvoltagesand currents).

hat,ifoneis interestedincomputingthe

ndcurrentinmerelya restrictedportionof

sible toreplacetheactualsourcesofexcitation

tagesource"orby asingle"equivalentcur-

ppropriatepoint.Thesetwospecialized

tinent,respectively,totheloopandnode

ThGvenin'sandNorton'stheorems.Basically

etransformationwherebytheneteffectof

dsourcesofbothtypesisreplacedby asingle

f interest.

ourceequivalenceisimpliedby thispro-


/ h t t p : / / w w w . h




N'STHEOREMS

we discusscarefullythereasoninginvolved

arlypresentedandanymisleadinginterpreta-

weareinterestedonlyinthe currentinone

excitedinarandommanner.Anadditional

tothisbranchcanbe adjustedtosuchavalue

meszero;thatistosay,the additionalvoltage

the flowofchargethroughthisbranch.

eactofremovingtheadditionalvoltagesource

cecanhavenoeffectuponanything,not

ossthisgap,whichis thesamewhetherthe

or not.Thus,whentheadditionalvoltage

aluethatresults inzerocurrentinthat

and,being idle,itspresenceorabsenceis

easoningwededucethefactthatthevoltage

edbyopeningany branchequalstheparticular

esourcethatrenders thenetcurrentinthat

ationtheadditivepropertyoflinear net-

antcurrentdue tothesimultaneouspresence

mputedthroughadditionof thecurrents

akenseparately,wecaninterpretthecondition

chinquestionasthesum oftwoequalbut

honeiscausedbythe originalrandomexcita-

therby theadditionallyinsertedvoltage

mesclearthatthecurrentproducedin this

tationmayalternatelybeproducedthrough

sourcevoltageequalinvaluetothe negative

sacrossa gapformedbyopeningthebranch

nisacting.Thisopen-circuitvoltage(the

e)is calledtheequivalentThiveninsource

heeffectofanyoriginal randomdistribution

asthecurrentin thisparticularbranchis

maybe suchthat,whenthecurrentinthe

onisrenderedzero throughopeningitor

ingvoltage,currentssimultaneouslybecome

hesofthe network.Inthiscasethesingle

cevoltagereplacestheeffectoftheoriginal

nforthesebranchesalso,andthusturns out

mofsourceequivalence.Itisimportantto

gleTheVeninsourcevoltageisequivalentto

ononlyforthe computationofthosebranch


/ h t t p : / / w w w . h




ONANDRELATEDTOPICS

owhentheparticularbranchinquestionis

enin'stheoremdealswitha restrictedkind

emaysaythatityields anoversimplified

lsourcedistribution,thatneverthelessgives

tina specificbranchornetworkportion.

hosenatwill, themethodcanactuallybe

ts,butonly throughfirstdeterminingasmany

evoltages(barringsimplificationsresulting

terofthe givenexcitation).It,therefore,

specialartifices,"butturnsout tobeoneof

andfrequentlyischosentoreplacethebasic

ticalaswellasi nnumericalwork.

hatisdual totheonejust given,wecansee

cingtheeffectof thegivendistributionof

eregardedasacurrent.Thus,an additional

arallelwitha givenbranchmaybeadjusted

voltagedropinthatbranchto becomezero.

nizethepossibilityofaccomplishingsucha

posingthatthevoltagein thisbranchis

ngashortcircuitacrossitand thenreplacing

entsourcehavingavalueequaltothe short-

tsourceandtheshortcircuitare inthis

theyprovidethesamecurrent;neithersupplies

rminalvoltage(branchvoltage)iszero.

ationtheadditivepropertyoflinear net-

antvoltagedueto thesimultaneouspresence

mputedthroughadditionof thevoltages

akenseparately,wecaninterpretthecondition

chinquestionasresultingfromthe sumoftwo

es,ofwhichoneis causedbytheoriginalran-

ne,andtheotherbythe additionallyinserted

sitbecomesclearthatthevoltageproduced

omexcitationmayalternatelybeproducedby

hasourcecurrentequal invaluetothenega-

rtcircuitplacedacrossthebranchwhile the

g.Thissinglesourcecurrentreplacesthe

omdistributionofsources,butonly sofaras

rbranchis concernedunlessthestructure

thevoltagesinotherbranchesbecomezero

onis short-circuited.Inthiscasethesingle

nttotheoriginal excitationwithrespectto

ell.


/ h t t p : / / w w w . h




N'STHEOREMS

miningasingle sourceequivalenttoany

nasNorton'stheorem.Furthercomments

alreadymadewith respecttoThdvenin's

supposeweconsiderthe networkshownin

esarevoltagesandtheparametervaluesarein

eatedbyThevenin'stheorem.

wnisthecurrentt» inthe4-ohmbranchas

plyTheVenin'stheoremtothecomputation

epistodeterminethe voltageeocatthegap

ranch,asshownin Fig.17.Notingthat

inationleadsusto thetwosubsidiaryproblems

ltageappropriatetotheproblemof Fig.16.

d e2.Thelatteris readilyseentobegivenby

olts(59)

,maybeexpressedas

e2-ohmbranch asindicatedintheleft-hand

g.17. Thiscurrentwemaylikewisecompute

ngunderlyingTheVenin'stheorem,andto this


/ h t t p : / / w w w . h




ONANDRELATEDTOPICS

the voltageeasshowninFig. 18.Byinspec-

ots

the terminalpaira-binFig.18 (thevoltage

edasshortcircuits)is

open-circuitvoltageofFig.17thevalue

hesinglevoltagesourceequivalentis

thissingleequivalentvoltagesourceisthe

itvoltage65and,

onoppositetothat

rowforeocin Fig.

vedthatthis

sourcewillyield

hecurrentu inthe

heveninsource

computecurrents

onemustfirst de-

gsinglesourceequiv-

entikin Fig.19maynowbeaccomplished

ardmanner.Forexample,onecandetermine

edattheterminalsof thevoltagesource(the

"looksinto"),whichis4 ohmsplusthenet

ecircuitofFig.16for thecomputationofthe


/ h t t p : / / w w w . h




N'STHEOREMS

eleft-handandright-handportionsofthe

2for theresistancebetweentheterminals

Fig.18,oneseesthatthe totalresistancein

4ohms(66)

s

amperes

etworkofFig. 16withtheoneof Fig.19,

that,whentheoriginalexcitationisregardedas

bothvoltageandcurrentsources.

edbecomeshortcircuitsbecausezero voltage

s pointedoutinArt.7 ofCh.2.5Kheft-eur-

edL-onemustbe remindedofthefactthat

tswhejatheir

oltage

eninFig.

aluesagain

cesareun-

thvalues

thvalues

parallelor

respective

thatin

eequiva-

gee,is thatattheopen-circuitedterminal

shownin Fig.21whichcontainsallof the

cevolt-

oblemin


/ h t t p : / / w w w . h




ONANDRELATEDTOPICS

inalnetwork.Asimplewaytofind this

ethevalueofthecirculatorycurrent*.To

hoffvoltage-lawequationsuffices,namely,

=0

ere

,is thevoltageacrossthe6-ohmresistor.

sistorin thedownwarddirectionis6+t

)

tworkofFig.21dead,thereis leftasimple

resistanceof3ohmsbetweenthe terminals

heTheVeninequivalentoftheoriginal net-

entofthecircuitofFig.20 forthecomputationofcur-

ightofa-a'.

atshownin Fig.22.Theportiontothe left

acestheactivepartoftheoriginal network,

thecurrentsinallof theremainingbranches

hmbranch

cuitof

eoriginal

henetwork

otted

'sthe-

stoobtain

ttothe

wecandojso

sivepor-

20asa

theshort-

becomesthatindicatedinFig.23 which

urrent

ofthecir-

theorem.


/ h t t p : / / w w w . h




N'STHEOREMS

veportionof theoriginalnetwork.Clearly

circuit,ratherthanacrossthe3-ohmbranch,

tagein theremainingpassivebrancheswould

edtohave zerovalues.

uationappliedtothenodetowhichthe

seentoread

0

herefore,is

valentofthecircuitof Fig.20becomesthat

equivalenceholdswithrespecttotheentire

ofthecircuitof Fig.20comparabletotheThevenin

erminalpaira-a',becauseshort-circuiting

currentsandvoltagesin thisportionzero.

showninFigs.22 and24revealstheequiva-

ee,= 42voltsinseries withthe3-ohmresist-

cei,= 14amperesinparallelwiththis resist-

seintherestof thenetworkisconcerned,a

followdirectlyfromthediscussioninArt.7

byProb.11of thatchapter).Thisequiva-

dcurrentsourcestogetherwiththeirassociated

dilydemonstratedinanalternatewaythrough

Norton'stheorem.Thuswe cansaythat

alentsofaras theireffectuponanexternal

heir open-circuitvoltagesandinternalresist-


/ h t t p : / / w w w . h




ANDRELATEDTOPICS

short-circuitcurrentsandinternalconductances

eor currentsourcetogetherwithitsasso-

ctanceishereregardedasasourceunit for

eisa functionoftheterminalcurrentorvice

towhichtheidea ofequivalencerefers.

rcearrangementsshownin Fig.25reveals

yequivalent,sincetherelationbetweenthe

rrentt<is thesameforboth,asthe reader

onofthegiven equivalencerelationse,=Ri,

avoltageintoan equivalentcurrentsourceandviceversa.

serelationsthatonemayregardany two

ftheyhave thesameterminalvoltageon

terminalcurrentonshort circuit.Reference

ver,showsthatthenetresistanceoftheactive

ncetotheleft ofa-a')isequalto theratio

etotheshort-circuitcurrent.Henceaknowl-

essufficestocharacterizetheactiveportion

ar asthedeterminationofvoltagesandcur-

reconcerned.

ersionsonemustbe mindfulofthestated

heequivalenceholdsonlyforcurrentsand

ernetworkportionstowhichthesourcesmay

eequivalencespecificallydoesnotapplyto

associatedsourceresistanceRor sourcecon-

eofthis artifice,letusreconsidertheprob-

cuitvoltagee,in thenetworkofFig.21.

alto theseveralsourcesinvolved,oneis

hesourcetransformationrelationsofFig.25

ocurrent sourcesintoequivalentvoltage


/ h t t p : / / w w w . h




N'STHEOREMS

dthe arrangementofFig.21asbeingequiva-

6.Herethecirculatorycurrentisseen tobe

viousone givenbyEq.69.Theterminal

undfrom therelation

75)

correct.

anconcludethatTheVenin'sandNorton's

ascomputationalaidsifthe givenexcitation

through-

smoreor

nwhichmay

of thenet-

ngleter-

heopen-

ircuitcur-

hensuffices

activeFiG. 26.'ThecircuitofFig.21 with

orkbyite currentsourcesconvertedinto

units shownvoltagesources,

tributionismore randomsothatitbecomes

ismannerseveralseparateactiveportions

tivesimplesourceunits,the computational

elydiminished.Itobviouslyceasestoexist

eunitsneededto replacethegivenexcitation

wointegersIorn characterizingthenumber

odebasis,forthe generalmethodsofanalysis

1 ofArt.8,Ch.2) thatanyrandomexcitation

hatnumberofequivalentsources.Infact,

tageofmakingasourcereplacementprobably

networkwithonlyamoderatelyrandomsource

ownbythe networkofFig.16,andonemust

ordertojudgebeforehandwhethertoapply

reinsuchacaseorto usethemethoddiscussed


/ h t t p : / / w w w . h




ANDRELATEDTOPICS

m

ofall linearpassivebilateralnetworksmay

tthe ratioofexcitationtoresponse,witha

tonepointand theresponseobservedatan-

terchangeofthepointsofexcitationandob-

erpretationofthisstatement,togetherwith

,providesthe topicfordiscussioninthis

e assumethatthenetworkisexcitedbytwo

bridgedacrossnodepairs towhichthevari-

ndregardallelementsasbeing pureresistances

ehavethroughoutthediscussionsofthis

quationsthenread

en= *,i

VQln^n=t,2

VQ3rfin=0(76)

hgnnen=0

thegivennetworkcouldberegardedasthose

gevariableseiande2,no restrictionisimplied

tionssofarasthe selectionofthesenodepairs

ever,makingthetacitassumptionthatthe

ilibriumequations(i.e.the Kirchhoffcurrent-

esameas thosewithwhichtherespective

ociated,sothat(accordingtothediscussionin

ermatrixfortheseequationsissymmetrical;

ulfilltheconditiong,k=g^,.

mplecaseforn= 2.Wethenhave

r ei,wefind


/ h t t p : / / w w w . h




REM

olvingfore2gives

theresult

nyieldingsymmetryfortheparameterma-

2 areassociatedwiththesameterminalpair

etheresultexpressedbyEq. 80maybe

tio ofvoltageresponsetocurrentexcitation

whichof thetwoterminalpairsis thepoint

hepoint atwhichtheresponseisbeing

ciprocitytheorem,maybeseento apply

gnowtotheEqs.76 pertainingtoanarbi-

yweshallshowthat,ifwesystematically

ethevariablesen,en-ii••-, e3,theremaining

ande2havethe formofEqs.77in whichthe

02iholds.Thatisto say,thesymmetryofthe

6is notlostthroughapplyingthesystematic

recognizethispropertyoftheequations,

atrix

quivalenttothe eliminationofen,asfollows:

sthrow,subtracttheg,n-multipliedlastrow,

hematrixofthe resultingequationsinvolving

en-ithenhasthe formindicatedby

nn— 0n201n)(0130nn~0n301n)'"'

nn~0n202n)(0230nn~0n302n)• 1"

nn— 0n203n)(0330nn-0n303n)1"'


G o o g l e - d

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_ u s e # p d - g

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ANDRELATEDTOPICS

ditiong,k=gk* pertainingtotheoriginal

etryofthematrix82.That istosay,the process

enfromEqs. 76leavesthesymmetryoftheir

nt.

inationofthevariableen-ifollowsthe same

ixfortheequationsinvolvingei •••e„—2

andsoforth.Henceweshall eventually

nsinvolvingeiand62 that,likeEqs.77,have

matrix;whencetheresultexpressedbyEq.80

emaywithcompletegenerality,therefore,

ylinearpassivebilateralnetwork,adetermina-

producedatoneterminalpairto thevalueofcur-

notherinvolvesnodistinctionbetweenthese

equallywellstartfroma setofequilibrium

s.Theexcitationthenis avoltageinserted

ponsea currentinanotherbranch.Inthis

tto voltageturnsouttobe independentof

containsthevoltagesource.

scommonlyusedtoprovethe reciprocity

compactformfortherelation betweenexcita-

dbytheuse ofdeterminantsinthesolutionof

hus,Eq.36in Art.2expressesaresponsex*

ntermsof thedeterminantAandcofactor

22.Theindexess andkmayreferto anytwo

mplied,orto anytwonodepairsif anodebasis

oneindex characterizesthepointofexcitation

thepointat whichtheresponseisobserved.

eterminantA,Eq.23,fulfillthe symmetry

eterminanttheoryshowsthatthecofactors

amelyA,k=Ak,.In Eq.36relatingexcita-

emaythereforeinterchangetheindexes

validityofthisequation;whereuponthees-

eoremisseento followatonce.

above,whichisbasedupon showingthat

ofagroupofequationsis invarianttothe

iceliminationprocedure,isnolessgeneral

erminants,andhastheadvantagethatit

braicbackgroundbeyondthatgainedfrom

ofnumericalexamples.

theoremitis significanttoobservethat

realwaysinvolved.Thus,ofthe twoquan-


G o o g l e - d

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_ u s e # p d - g

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REM

onse,onemustbea voltageandtheothera

eryspecialcases,reciprocitydoesnotapply

heresponseareboth voltagesorbothcurrents.

onis easytosee.Thus,supposebothquan-

tagesource(ashas frequentlybeenpointed

tcircuit,whiletheobservationofavoltage

pairimpliesan open-circuitcondition.Hence,

ort-circuitedwhilethepoint ofobservationis

hangeofthepointsofexcitationandobserva-

nterchangeofopen-andshort-circuitcon-

airs,and onecannotunderthesecircum-

uationtowhichthereciprocitytheoremapplies.

responsetoexcitationto remainunchanged

ryisaltered.Althoughonemayfindsome

procityneverthelessstillholds,itis ingeneral

itationandresponsearebothcurrents,fora

lyanopencircuitwhiletheobservationofa

airimpliesa short-circuitcondition.Hence,

tsof excitationandobservationagainnecessi-

en-andshort-circuitconstraintsatboth

tageand theresponseacurrent,then

reimpliedatbothterminalpairs;and,when

nd theresponseavoltage,theimpliedcon-

uits.Hence,inthesecases nochangeinthe

mpaniesshiftingthesourcefromone terminal

ethesituationstowhichreciprocityapplies

henetworkshownin Fig.27,where,for

umbersonthebranchesmaybe regarded

stancevaluesin ohms.Usingtheprocedure

orks,asdiscussedinArt.3,one obtainsfor


/ h t t p : / / w w w . h




ONANDRELATEDTOPICS

Fig.27

ts

mperes

/8volts

fexcitationandobservationyieldsthe

27,forwhichwehavethe computational

es

s

amperes

59/8volts

withtheresult 84.

uationtowhichthereciprocitytheoremdoesnotapply.

iprocitydoesnothold forthissamenet-

age-to-voltageratio.Figure28illustrates

tryinvolved.Forpart(a)wehavethecom-


/ h t t p : / / w w w . h




NSFERFUNCTIONS153

peres

lts

)ofFig.28,on theotherhand,wehave

eres

volts

doesnotholdwithrespecttothe ratioof

vefromcomputations87thatfa/ji)= 8/9,

that(ei/j's)=8/9also. Thesearesituations

applies,forjiand j5mayberegardedas the

ources.

ron,reciprocityis anetworkpropertythat

telyitisfrequentlymisused,forthe restric-

enot alwaysclearlyobservedbythosewho

acticalproblems.Thestudentshouldreread

entofthereciprocitytheoremasit isgivenin

hisarticleand besurethathe cansupply

provisionstoestablishitona foolproofbasis.

erFunctions

quilibriumequationsontheloop basisfora

glevoltagesourceis present.Ifthepointof


/ h t t p : / / w w w . h




ONANDRELATEDTOPICS

thesth loop,theequationshavetheform

nt

olutionforthe currentinloopkmay,accord-

t.2(Eq.36),

et-

sidering

ess.

erpreta-

pposethat

areidentified

ennetwork,

n having

ythrough

rkiscom-

asshown

helinks

as tohave

yofshort-cir-

seterminalpairsarespokenof asthepoints

asitsexternalterminalpairs.The net-

I terminalpair.* Theterminalpairsarethe

sibility.

nfor anetworkhavingaspecifiednumberofterminal

houldbeobservedthatthere isnohyphenbetweenthe

ove)andtheword"terminal"becausethisformwould

I terminals,whichisnotthecase.Properlyspeaking,

anyterminals;it hasonlyterminalpairs,andthere are


/ h t t p : / / w w w . h




NSFERFUNCTIONS155

tationischosenasbeing inloop1or at

quallywell belocatedatanyotherterminal

nalpairsare short-circuited,eventheone

cated,forthe voltagesourceisageneralized

ressedbyEq.93,the quantities

nsoftheintegersk andsareregardedas

thecurrentresponseatanyterminalpairis

ingtheexcitationbythisfunctionforappro-

hepresentinstancetheexcitationis a

sa current,sothattheresponsefunctiony^,

oltage,andisdenotedas anadmittance.

relates currentandvoltageatthesame

driving-pointadmittance,sincethepointof

oas thedrivingpoint.Fors ^k,thequan-

radmittances.Multipliedbytheexcitation

rs, thequantity2/*,yieldstheshort-circuit

k, allotherterminalpairsbeing simul-

.Thecompletesetofyk,are,therefore,spoken

ng-pointandtransferadmittancesoftheI

emayconsiderthe equilibriumequations

followingonesonthenodebasis:

0

,(95)

=0

ourceis assumedtoexcitethenetwork,and

depair.In termsofthedeterminantmethod

orany node-pairvoltage


/ h t t p : / / w w w . h




ONANDRELATEDTOPICS

Eqs.95is

kewisethinkofthenetworkasbeingen-

nFig.30,withterminalpairs broughtout

chthevoltagese\•• •enrefer.Incontrast

.29, theterminalpairsinthe presentrepre-

uited,even

t,forthe

zedopencir-

net-

sidering

faccess.

sinthis case,

eatanytermi-

tiplying

ppropriate

we refer

-pointfunc-

ferredto

eitrepre-

current,it

sbeingthe

ucharatio). Multipliedbytheexcitation

s, thequantityz*,yieldstheopen-circuit

rk, allotherterminalpairsbeingsimulta-

hecompletesetofz*,are,therefore,spoken

ng-pointandtransferimpedancesofthen ter-

aybeseento havethedimensionsof

ynotingthatthetermsin thecompleteexpan-

elementsr,* whilethoseintheexpansionof

suchelements.Inlikemanner,thez*, of

vethedimensionsofohms(reciprocalmhos)by

ecompleteexpansionofGareproductsofn

ntheexpansionof areproductsof(n— 1)

dimensionallythez*.andyk,arereciprocal,


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NSFERFUNCTIONS157

attheyarenumericallyreciprocal,asatten-

ationwillreadilyreveal.

nantsthat,iftheelements,likether,* in

nditionr,* =r*„thensodothecofactors;

equilibriumEqs. 91fortheloop basis

havesymmetricalmatrices,thenwehave

venaboveforthesequantities,werecognize

erificationofthenetworkpropertyreferred

emandsubstantiatedmorecarefullyinthe

restedinthe currentsinalltheloops ofa

at allofits nodepairs.Specifically,with

e networkrepresentationshowninFig.29,

yin regardingsayii,i2,• ••,ipas accessible

tionofFig.29 thissituationisreadilytaken

ytheterminalpairs1 •••P, whileanalytically

byapplyingtoEqs.91 thesystematicelimina-

minatethevariablesip+i••• *'i.Theresulting

iningvariablesii• ••ipare thenrelatedtoa

pairnetworkinthesamefashionthatEqs.91

al-pairnetworkofFig.29.

dgethenterminal-pairnetworkofFig.30

yeliminatingfromEqs.95thosevariables

alledfor.

et ofad-

z,* for

ofaccess

chosenasto

rtherdetail

unteredsitua-

henumber

Sucha so-

etworkisrepresentedbythesketchinFig. 31.

weenthecurrentsandvoltagesmaybere-

oma setofnodeequationslike95 written

urcesi'iand i2,orfroma setofloop

ortwononzerovoltagesourcesei ande2.

he firsttwodependentvariables,oneob-

rnet-

tionalcur-

edirec-


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ANDRELATEDTOPICS

eitherin theform

nversepairsof equationssincetheypertain

anceattheterminalpair 1whentermi-

edasmaybeseenfromthefirst Eq.100by

fore2=0. Similarly,y22istheadmittance

minalpair1short-circuited.Theshort-

ey2iisina likemannerinterpretedasthe

uitedterminalpair2,toanexcitationei at

theratioofii attheshort-circuitedterminal

appliedtotheterminalpair 2.Bythereci-

oratiosareequal(yi2 =y2i).

emayinterpretphysicallythez,u in

2areseentobe theimpedancesatthetermi-

velyunderopen-circuitconditions.Forexam-

vealszuto beequaltoei/*'1for i2=0 (open-

alpair2).Similarly,this sameequation

or ii=0;thatis tosay,itequalsthe volt-

circuitterminals1-1'perampereof current

2'.The impedancez2iisinterpretedina

ndofEqs.101 astheopen-circuitvoltage

,per ampereofcurrentappliedtoterminal

oremagainlendsphysicalsignificanceto

nterpretationstospecificexamples(asis

e readeriscautionedtogivecarefulatten-

sshownin Fig.31,whicharechosenina

stodealwiththe questionof"input"and

nner.

oobtainrelationslikethose inEqs.101,

ctionbetweenthez'sandy's,


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SFERFUNCTIONS

heEqs.100 isdenotedby

12=2/21beingtacitly assumed.Thecon-

y'sin termsofthez's,thus

=-r^r (104)

Eqs.101is

)

onz12=Z2i holds.Thedeterminants|y\

ciprocalvalues;thatis,

ma comparisonofrelations102and104.

terestinganduseful relationshipexpressed

ratethe unusualcharacterofthisresult.

orkof

entvaluesj.0WWtQ2*

znisthe

mbination

ywis the^8

branch

iQ32. asimpleexampleofadis-

twoterminal-pairnetwork

two/.

expressedby

hatz22 isEq.107is illustrated,

ncealone

ceof thetwobranchesinparallel.Thus we

ythepairsofquantities108 and109seem

attheydofulfillthe conditionexpressedby

st,since thisrelationshipholdsforanytwo


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ANDRELATEDTOPICS

t isoftenusefultoexpressthe quantities

viceversa.Suchrelationsarereadilyobtained

anipulationofEqs.100or 101.Itiscus-

erminethecoefficientsA,B,C,D,called

eters,intermsofthe y'sorthez's. Thefol-

xplanatory,andmakeuseofEqs.100,101,

tingthey'sandz's.

hedeterminantofEqs.110is seentohave

onsbecome

s(likethethree y'sorthethree z's)areneeded

minalpair,it istobeexpectedthatthe four

tbearsome relationtooneanother.This

esimpleoneexpressedbyEq.115;and, asa

16, inversetoEqs.110,havecoefficients

uantities A,B,C,D,andthat thesetwoin-

alinformexceptfor aninterchangeinthe

entsAandD.

serelationsinvolvingtheparameters

pressingtheinput impedanceZiofatwo


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FIGURATIONS161

.33)intermsofits loadimpedanceZ2.Since

2),wherethevoltageandcurrentreference

oted,weseethatdivisionofthe firstofthe

e yields

theory,thisrelationshipbetweenthequan-

salinear

.Physically

Fig.

Z2into

es of

ksinprac-

nsforma-

yuses,

equentlyinlaterdiscussions.

ngim-

17.

gurationsandTheirEquivalence

tionsfortwoterminal-pairnetworksoccur

quentdiscussionstojustifytheirindividual

norder

terizations

smayap-

d.

edtee

xes

heir own

edances

orks.That

ontainsingleelementsoranygeometrical

havingoneaccessibleterminalpair.The

andtransferimpedancesofthetwotermi-

histeeare seenbyinspectiontobe

YaYc(118)

YbYc(119)

eadmittancescorrespondingtotheimped-

.


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seentobe

ZaZb+ZaZc+ZbZc

104,theshort-circuitdriving-pointandtrans-

etheseresultswiththosefoundforthe

ownin Fig.35,whichismoreappropriately

dmittances

terms

ancesY\,

nby

pressions

-point

character-

network

usingthe

they,kgivenin theparagraphfollowing

eviousarticle,andnotingwellthe reference

.

3 (125)

3)/Z2Z3(126)

impedancescorrespondingtotheadmit-

een tobe


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FIGURATIONS

102,theopen-circuitdriving-pointandtrans-

mepi networkbecome

3, «v

+Z3Z1+Z2+Z3

OA»

+Z3Z1+ Z2+Z3

teeandpistructuresshould beratherclear

erelationshipscharacterizingthem.The

ecomeequivalentarereadilyseenfrom an

sgivingtherespectivey,kand z,*.Thus,

118,119, 120withEqs.129,130,131 wesee

tee intermsofthoseof thepiaregivenby

122,123, 124withEqs.125,126,127 yields

fortheadmittancesofthepi intermsofthe

Yc

uationisinevidence,sincethe transforma-

2)onan impedancebasisisidenticalin form

omteetopi (Eqs.133)onanadmittance

alsothatthe teeandpinetworksdiscussed

meas thewyeanddeltaconfigurations

Art.4,and thatthedelta-wyetransforma-

nttothepi-teetransformationEqs.132above,

ormationEqs.51areequivalenttothetee-pi

Thusthephrase"productofthe twoadja-


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ANDRELATEDTOPICS

allthree"isseen tobeacompactwayof

altothe resultsgivenbyEqs.132and 133.

stothese remarksthereadershouldmake

andpinetworksthatare analogoustoFigs.9

gtheabovephraseto obtainEqs.132and

he pertinentsumsZi+Z2+ Z3and

plephysicalsignificanceofbeing,respec-

earoundthemeshofthe pi(Fig.35)andthe

tinthecentralnodeofthe tee(Fig.34),

sumsYi+Y2 +Y3andZa+ Z0+Zc

attice(b)andits "bridge"circuitequivalent(a).

slybetemptedtouse)have nosuchsimple

hreferencetothecircuitsinvolved.

,Za=Z0lor, inthepiof Fig.35,l^i=Y2,

be symmetricalwithrespecttotheirinput

foran interchangeoftheseterminalpairs

the electricalbehaviorofthelargernetwork

beembedded.Asymmetricalstructure

portantpartinnetworktheoryis thelattice

.Inpart(a) ofthisfigurethenetworkis

miliarformofa Wheatstonebridge;inpart

rawnasalattice,usingthe conventionof

rthanrepeatingtheboxes zaandz0.

equivalenceofthebridgeandthelattice,

takingholdof thelowerright-handboxza

tover(endfor end)longitudinallysothat

chother.The terminals1-1'willthenbe at

andtheterminals2-2'will beattheupper

ofthelatticeare thussometimesreferredto

e impedancesz0asthe"cross-arms."Notice,


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FIGURATIONS

seitherterminalpair 1-1'or2-2'(not both),

crossedand theZ\,boxesbecomeuncrossed.

ticehasthepropertythat aninterchangeof

szaandz<,is equivalenttoatwistofeither

thelattice,unliketheteeor pinetworks

sessymmetrywithrespecttoahorizontalcen-

dwithitssymmetryaboutaverticalcenter-

ywithrespectto thetwoterminalpairs).

eis usuallytakentobeat areferenceor

metricallatticeisspokenofas being"bal-

nd."Thesymmetricalteeor pistructurein

being"unbalancedwithrespecttoground"or

nbalancednetwork."Thustheterms"bal-

areusedtoreferto symmetryorassymmetry

ne,whiletheterms"symmetrical"and"dis-

nconjunctionwithatwoterminalpair, refer

metryaboutavertical centerline.

ointandtransferimpedancesofthesym-

e readilyfoundbyinspection.Thus,with

tethatthe circuitbetween1and1' consists

beinga seriesconnectionofzaand zj.Hence

4)

ittransferimpedanceaccordingtothe phys-

nthepreviousarticle, letussupposethat

erminal1andwithdrawnfromterminal1'.

theresultingvoltagerisefromterminal2' to

two parallelpathsjoiningterminals1-1'

t\amperetraverseseach path.Either

szaina directionconfluentwiththecurrent

ndzj inacounterfluentdirection(yieldinga

ve

clearthatz12=0 forz0=z&, whichisrecog-

whichthebridgeis balanced.

thezdeterminantisreadilyseen tohave

136)


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ANDRELATEDTOPICS

rtheshort-circuitdriving-pointandtransfer

137)

/zbarethe latticeadmittances.These

interchangeoftheletters yandz,identical

and135,thusshowingthat thesymmetrical

quivalencerelationsbetweenthelattice

ig.34withZa =Zb)bycomparisonof

s. 134and135.Thuswesee that

za)(139)

2ZC(140)

eequivalencerelationsbetweenthelattice

g.35with Yi=F2)by comparisonofEqs.

7 and138.Thuswehave

(141)

S(142)

presentingthe conversionofalatticetoan

re)containminussigns, itisclearthat a

ositiveresistancesinitsbranches)doesnot

transformationviathe lattice.Elementvaluesare

sicalequivalentteeorpi.A physicalsym-

therhand,alwayspossessesaphysicalequiva-

ersionrelations140and142 involveonly

gtheseequivalencerelationsisshownin

isthe symmetricalteeattheleft.Through


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FIGURATIONS

yconvertedintothelattice,andthe latteris

e picircuitontheright accordingtothecon-

ngthatonemust nowreasonintermsof

veadmittances).

teseriesoftransformationsisshownin

theso-called"bridgedT"attheleft,

eeofFig.37 witha20-ohmresistancebridged

p intheconversionprocessconsistsof

asa parallelconnectionoftwonetworks,one

othermightbe regardedasapi thathaslost

ticetransformationviatheparallel combinationof

tvaluesareinohms.

nextstep,eachnetworkistransformedtoa

eistransformedintoa latticeandthe20-ohm

-ohmresistorsinseries.Thelatter network

cethathaslostits cross-arms.Inthefinal

againcombinedintoone,this beingagaina

resultantlatticeisobviouslytheparallel

-ohmresistances,andthezjarm isthesame

ticeofthepreviousstep.

ridgedtee toanequivalentlatticeis

esubsequentconversionofthelatticeinto a

esired,bedoneas inthepreviousexample.

thereversedirection,onemayconvertfrom.

and thisconversioncanbedonewithmany

onofoneresistanceintotwoparallel ones

infinitevarietyofways.

exibilityishadthroughnotingthefol-

tice.Equations140showthat,if zaandzj

erm,thenZaofthe equivalentsymmetrical

also.SinceZaisin serieswiththeinputand

hatanycommonimpedanceinserieswithza


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ANDRELATEDTOPICS

egardedas beinginserieswithboth theinput

resultis illustratedinFig.39.

wthat,ifyaandyb haveacommonaddi-

uivalentsymmetricalpihasthisadditive

latticeproperty.

rallelwiththeinput andoutputterminals,

dmittanceinparallelwithya andybmay

beingin parallelwithboththeinputand

ltisshownin Fig.40.

ese latticeproperties.Fromtheoneshown

atelyobtaintheequivalenttee.Thusinthe

poseweregardthe entirezaarmasthe com-

ovalasin Fig.39leavesaremaininglattice

eries armsandanimpedance(z&— za)for

of thepropertyshowninFig.39.

twocross-armsarethusplacedinparallel,

tingstructureis ateewithits seriesand

yEq.139.Inan analogousmanneronecan

ig. 40toobtainatonce theequivalentpi

ractionofthe givenlatticeresistancesor

onadditiveterm,andapplyingtheproperties

40alternatelyinacontinuingsequence,one

oanequivalentladder.Figure41 showsan

herethemethodofFig. 39isfirstapplied,and


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RANSFORMATIONS

tocompletethedevelopment(whichcould

roughalargernumberof steps,ifdesired).

hisprocesstoconvertasymmetricalladder

enceintoanequivalentteeorpi.

entofthelattice,usingthepropertiesshownin Figs.39

ein ohms.

edecompositionofasinglelatticeinto the

o,andthesubsequenttransformationofthese

Fig.42.Theresultantstructureonthe

parallel,anarrangementthatiscommonly

helatticeisreadilyconvertibleintoa twin

transformation.Elementvaluesareinohms.

quivalentlattice,throughuseofartificesof

ssing.Infact,anendlessvarietyofaddi-

ereadilyobtainable,asthereadermay now

nsformationsunderWhichThey

ohmsconductsacurrentofi amperes,the

sRise= Rivolts,andthepowerdelivered

ergydissipationin R,is

persecondor watts(143)

powermaybethe voltagee(animpedance-

tmay bethecurrenti(an admittanceless


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ONANDRELATEDTOPICS

pression143forthepowerdeliveredor con-

ase.Significantis theresultthat,ifthe

wervaries inverselyastheresistance(thatis,

tanceG=l/R),whereas,ifthesourceis a

proportionaltotheresistance.Inboth cases

othe squareofthesourceintensity.

e, i,aknowledgeofanytwouniquely

learfromtheconsiderationthattherearetwo

ingthesefourquantities,viz.:P=ei, and

randominterconnectionofvariousresist-

thetotalpowerconsumedinseveraldifferent

epowercalculationsgivenbyEqs. 144through148

uesareinohms.

rstthe currentsinalltheresistancebranches,

achbranchusing therelationP=t*R,and

y addingtogethertheresultsforthe separate

atelyfindfirstallofthe branchvoltagesand

orbedbyeachbranch;and thereareobvi-

oftheseschemesforcomputingseparately

onepoint only,itmaybesimplerto com-

ceattheterminalsofthissourceand then

at suppliedtothisnet resistanceasthough

placedbyasingle resistancehavingthis

dnetresistanceordriving-pointresistance

yasaresistanceofsuch avaluethatthe

esourcebecomesidenticalwiththatabsorbed

ethecomputationsinvolvedinthedeter-

ntresistanceareusuallyaboutasextensive

omputationofthevoltagesorcurrentsin

enetwork,thisalternativewayoffinding the

ilycomputationallysimpler.Experience

dicatethebestmethodto useinagiven


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RANSFORMATIONS171

otenoughdifferencetorenderthe decision

sillustrate withasimpleexample.

ssumedtobedrivenat theleft-handend.

ebeginbyassuming,fore4,1volt.Then the

culationsareself-evident.

ere

ythe sourcemaybecalculatedbyadding

bedbythe separatebranches.Sinceeither

achbranchis knownfromcalculations144,

ybranchis obtainedbyuseoftheappro-

143. Beginningattheright-handend,this

g

+2+(25/5)+10X4=50watts(145)

ving-pointresistanceas

hms(146)

werasthe singleterm

X2=50 watts(147)

rrent

watts(148)

1voltinsteadof25 volts.Thenallthe

ecalculations144areHsth aslarge,all

thaslarge,andthenet powerinputisM25th

ntheotherhand,if etweregivenas

argerthanitturns outtobein thecalcula-

nthesecalculationsare4timeslarger,and

ybranchaswell asthetotalpoweris 16times


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onedrivingpointinthe network,corre-

ousapplicationofseveralsources,thenone

appropriateextensionof theabovepro-

sin anycaseexpressibleasasumof the

bytheseparatebranches,andthe power

mputedfromthepertinent branchcurrent

me way,irrespectiveofwhetherthesecur-

sedbya singlesourceorbyseveralsources

owever,whenseveralsourcesarepresent,we

vetocomputethecurrentorvoltageina par-

ngthecontributionsduetoeachsource

ngthesetogetthe netvalue.Fornsources,

orthecurrentin aparticularbranchan

(ii+*2H— •+tn)whereeachterm isthe

sourceactingalone.If thepertinentbranch

thenet powerabsorbedbythatbranchis

*R(149)

•••+2iiin(150)

atwhatwecannotdo,butmightbe tempted

powerabsorbedbythisbranchthroughadding

erdissipatedin itowingtothecurrentcom-

kenseparately.Thissum,whichisequalto

. .+inevidentlyfalls shortofthecor-

tributionscomingfromthedoubleproduct

mesclearthatwecannotcomputethenet

fedfromseveralsourcesby computingthe

ourceseparatelyandaddingthese.Therule

positionthatapplies tothecomputationof

otapplyto thecomputationofpower!

sonforthis conclusionfromseveraladdi-

f all,weshouldberemindedofthe fact

opertyappliesonlyso longastheanalytic

ar.Onlysystemsorsituationsgovernedby

dditivepropertyorpermitthe superposition

expressingpower,asgivenabove,arequad-


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TRANSFORMATIONS

should expectthesuperposabilityofsolu-

precedingdiscussionsshow.

helpfulhere.Consideringtheseveral

ertainlybetrue thatthenetpowersupplied

ousvaluesofpowersuppliedbytheseparate

hisstatementis correctonlyifwec ompute

hsource whilealltheothersare simultaneously

ourcein questionaloneisacting.Theim-

mountofpowerthatagivensource supplies

erminalvoltageandcurrent;ifthesource is

dependsnotonlyuponthisvoltage,butin

ntensitiesof theothersources,and,ifthe

svoltagedependsjointlyuponthe intensities

epowersuppliedbyanindividualsource

towhetherit isactingaloneor inthepres-

erat thesamepointorat otherpointsinthe

uationofthissort istocomputethenet

thesources,usingsuperpositiontodothis

epropertydoesapplytothecalculationof

mthenetvaluesof sourcevoltageandcur-

et powersuppliedbyeachsourceandthrough

wer.Alternatelywecanc omputethenet

separatebranches,againusingforthispur-

positionif wefinditexpedientto doso.From

ecan thenreadilycomputethenetpower

ndthroughadditionget thetotalpower.

orthetotal powerisobtainedbystarting

tionsontheloop basis

e2

opcurrents,ei

ps,and

s


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ermatrixonthe loopbasis.Sincethetotal

workisgivenby

3)

orbedby thenetworkisobtainedthrough

essivelybyt"i,i2,•• .,iiandaddingthe results.

i

54)

omogeneousandquadraticinthevariables

adraticform,and[R]inEq. 152isthematrix

erwhatrealvalues(positiveor negative)

variablest'i •••ij, thisquadraticformmust,

e powerabsorbedbyapassivenetwork

draticformhavingthispropertyis called

fficientmatrix[R]arisesfroma givenpas-

yielda positivedefinitequadraticform;

sspecifiedand acorrespondingpassivenet-

esisproblem),thenasolutioncanexistonly

tivedefinitequadraticform.Lateron,

edwiththiskind ofproblem,wewillshowhow

testedtoseewhetherit meetsthiscondition

ehandwhetherornotacorrespondingpassive

rrelation154isthus seentobeusefulin

atofaffordinga meansforcomputingpower.

eadilybeobtainedfromexpression154.

rloopcurrentt'* isafactorof thefcthrow

s containedinnoothertermsofthis expres-

ueofPis unaffectedifwereplacethevariable

plythekthrowandthe fcthcolumnin[R]by

herway,we cansaythat,iftheparameter

ughhavingits kthrowandcolumnmultiplied

currenti* isdividedbythis samefactor,

rthe totalpowerremainsunchanged.Refer-

thepowerrevealsthatthesemanipulations

acementofe* by


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sandcurrentsinthenetworkremainunaltered.

voltagesource,thenachangeinthenet-

ondingtothemultiplicationofthefcthrow

ctoraleavesthepowerrelationsinvariantand

thevoltagesorcurrentsexceptthatthecurrent

dbythefactor1/aasexpressedby Eq.155.

be seendirectlyfromequilibriumEqs.

ltiplyalltermsin thefcthequationbythe

gthetransformationsthatleavepowerrelationsinvar-

.Elementvaluesare inohms.

oefficientsinthe fcthcolumnbya,andreplace

elast twooperationscanceleachother

ultiplyingalltermsin thefcthcolumnfirst

ethemultiplicationofanyequationby a

ectuponthesolutions.However,thefcth

multipliedbya,andonlythe voltageandcur-

fectedasshownbyEqs.155 and156.Since

tthe powerrelationshipshavenotbeen

napplytheseresultsin specificsituations.

43is redrawnindicatingthemeshcurrents

ver,areevidentlyidenticalwiththelikenum-

wninFig.43).Byinspectionone maywrite

etermatrix

ons


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ltisreadily foundtoyield

1/50ampere(159)

hesecondrowandcolumnof [R]bythe

e following

pondsthesamecircuitconfigurationas

theelementvaluesshownin Fig.45,asthe

gthecircuitof Fig.44toaninternal impedancelevel

dbythematrixEq.160.Elementvaluesarein ohms.

byinspection.Solutionofthecorresponding

ei=1 voltisseento give

)i2,i3 =1/50ampere(161)

angedbyafactor2/3.Thedriving-pointand

3 arethesameforthe circuitofFig.45as

ninFig.44.Thetransferratio ei/i*2is3/2

agesourceintomesh2,withthe valuee2

thevaluee*2 =(3/2)e2in thecircuitof

2Aiande*2/i3wouldturnoutto be3/2times

spectively,whiletheratio e*2/i*2wouldbe

.

f driving-pointandtransferimpedances

ircuitsexceptthatthedriving-pointimpedance

slargein thecircuitofFig.45,andany trans-

therthecurrentorthevoltageof mesh2

timesaslarge.Theoperationupon the

edoutintransformation160is, therefore,

mpedancelevelofmesh2bythe factor(3/2)2;

ance-leveltransformation.Itseffectuponthe

nvolvedis readilyseen,asinthe above

evoltageandcurrentrelationsin thecir-


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rehand,anditseffectuponthe powerrela-

posewedropthe impedancelevelofmesh3

hefactor(2/3)2as indicatedinthefollowing

ancematrix

owninFig. 46.Thecorrespondingequi-

heimpedancelevelin mesh3ofthecircuitin Fig.44by

gtothematrixmanipulation162.Elementvaluesare

orei= 1voltyieldthe followingsolutions,

/100 =(3/2)t3ampere(163)

erify.The finalresistorattheright ofthe

rdedasaload) is(8/9)=(2/3)2 X2ohms,

ngoftheimpedancelevelatthe outputby

geacrossthisloadis now(2/3)timesits

eliveredtotheload isunchanged.

emonstrateforhimself,onecannotchange

esiredfactorswithoutrunningtheriskof

etworksinvolvingsomenegativeresistance

earestillpassivenetworks,fortheyhave

ipsastheoriginal one,buttheyarenotphysi-

rms.Therefore,oneshouldavoidusing

eadtosuchresults.

nd picircuitsdiscussedintheprevious

nizethatadissymmetricalteeorpi isequiva-

withan impedancetransformationapplied

Statedinanotherway,wecansay: Except

nsformation,adissymmetricalteeor picir-

anasymmetricalone.f

elytrueonly forresistancecircuits.Theextentto

willbeseenin laterdiscussions.


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NDRELATEDTOPICS

edissymmetricalteeofpart(a)in Fig.47.

trix wecanignorethepresenceof the40-ohm

altee(a)subjectedto atransformationloweringits

thefactor1 /4yieldsthesymmetricaltee(b)with appro-

tance.Elementvaluesareinohms.

rcuit).Thuswe have

symmetricalteewill resultifwedepress

outputby thefactor60/15=4, asisdone

n part(b)ofFig.47 inwhichtheappropriate

40=10ohms.For equalinputvoltages,

ssthe10-ohmloadis1/2 thevoltageacross

rdeliveredtothe loadisthesamefor both

teresttonotethatwecan alsomakewye-

ionswiththismethod.Thisis moreeasily

od,andso inpart(a)of Fig.48wehavere-

indicatingtheelementvaluesinmhos,

des,andemphasizingthechoiceofdatum

nentnode-conductancematrixis

w andcolumnbyafactor whichisthe

diagonalelementstothediagonalelement

1/16),thenthe resultingmatrixhas

alterm equalsthesumofthe nondiagonal


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thefollowing:

rkisthatshowninpart (b)ofFig.48(ele-

hichthesignificantfeatureisthat thereis

7777,

77777)

)transformationeffectedthroughuseofthe impedance-

nique.Elementvaluesinparts(a)and(b) areexpressed

eexpressedinohms.

e2with thedatum.Changingtoresistance

ingtheseries branchesyieldsthepicircuit

re.

nsformationisbestdoneon aloopbasis,

himselfasanexercise.Thismethodof

mationshasnoparticularadvantagebutis

heseimpedance-leveltransformationscan

rcuits involvinginductanceandcapacitance

nts,andtheir usefulnessinprovidingasimple

elementvalueswithoutaffectingcertain

ationswill befoundtogiveus aninvaluable

thesis.Intheimmediatelyfollowingchap-

uaintthereaderwiththemost important

pacitiveelementsuponcircuit propertiesand

tionsbythesystematiceliminationmethod:

r2+5x»+6x4=33

+10z2+7x3-8x4=5

x2+2x3-10x4=-15

8


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ONANDRELATEDTOPICS

on,solvethefollowingthreeequationsets:

4n+12x2+3x3-0, 14ii+12x2+3xs-0

2x i+ llx2+3x3=1, 12x i+ llxj+3x3=0

3x2+x»=0, 3x i+3x2+xs-1

pressionsfor thex'sinthe moregeneralequations:

obs.1 and2bymeansof determinants.

ematiceliminationtechnique,reducethefollowingma-

ndthus evaluatetheirdeterminants:

networkscomputethevoltageand currentdistribu-

tdotheanswersbecomeif thesourceisa currentof

edladderin sketch(b)isequivalenttothe balanced

asthedeterminationofmeshcurrents isconcerned(since

eresistanceparametermatrixona meshbasis).Byfirst

tsinthe unbalancedladderpertinenttohaving1voltat P u b l i c D o m a i n ,

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alsof allthenodesin thebalancedladderinsketch(a)

alandnodem hasapotentialof100 volts.

=xvolts. Then,startingattheright-handend,find

nallbranchesandthevoltagesat allnodesexceptnode1

eequations

enfindallthevoltagesand currentsexplicitly,including

takei,= 1ampere,andreviseallvoltagesandcurrents

argrid shownhere,eachbranchisaresistanceof1 ohm.

nchesandthepotentialsofallnodesas wellasthenet

ampereisappliedat thefollowingterminalpairs:(a)nodes

ndc joinedtod, (c)ojoinedtodand6joinedtoc , (d)

ditionsandtheresultsofProb.7 toget(a);thencon-

assuperpositionsof appropriateonesoftype(a).


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ANDRELATEDTOPICS

hefollowingsketches,eachbranchisaresistanceof

edatoneof thepointsa,orb, determinethevoltages

ranches,first,consideringthesourcetobea voltage,and,

ea current.

tedat pointa,notethatsymmetrypermitsoneto

twoidenticalbalancedladderswhichmaybedealtwith

rceinsertedatpoint 6,itisexpedientto divideitinto

cesuchidenticalhalvesopposingeachother(equalto

e superpositionandsymmetrysoastoobtainthe desired

nganythingmorecomplexthanaladdernetwork.

rcuitshownhere (inwhichthecircularconductoriaa

byindependentreasoningandthenbyuse ofthestar-

mesh,and,inviewof thepositionofthevoltagesource,

es.

stances

tar-deltatransformations,reducethecircuitshownin

alancedladder,andsolvefor theratioe2/t'i.


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owin whichallbranchesare1-ohmresistances,solve

ansformationsappropriatetothesuppressionofnodes

sultingcircuitintheform ofair.

ductancematrixonanode-to-datumbasisandsys-

perationsanalogoustothoseexpressedbyEqs.49corre-

eeliminationofnodes2,3, 4.Thusbeginbyaddingto

4,5therespectivea-multipliedelementsofthesecondrow

elementsincolumn2 excepttheoneontheprincipal

e2 isdecoupledfromtherestof thenetworkandrow2

matrixas mayalsotheremainingcolumn2whichconsists

avea symmetricalfour-by-fourmatrixwhosecolumns

odepotentialseii,ej, e«,ef,.Treatthismatrixinprecisely

escribedfor theoriginalone,andobtaina symmetrical

osecolumnsrelaterespectivelytothenode potentialsei,

-by-twomatrixappropriatetothepi networkfoundby

heck.

rcuitof Prob.8,useThevenin'stheoremtocompute

mresistorplacedacrossnodes b-cwhen1voltis applied

orplacedacrossb-dwiththevoltageappliedacrossa-c.

e,the quantitiesdenotedbyra,n,•• •arethenet

dobtainatterminalpairs createdbycutting(aswitha

ectivemeshesinsuccession,eachtimeleavingtheprevious


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ANDRELATEDTOPICS

ningintermsofTheVenin'stheorem,showthat

obefed intoend6and thevoltagetobeobservedat end1;

ameexpressionfor ei/ij,thusprovingthereciprocity

ks.

edualtothesituationin Prob.14.Writethedual

arrythroughitssolution.

ecircuitofProb.14 are1-ohmresistances,obtain

or allthesignificantratios.

6inthetext writethemeshequations,andsolvefor

celimination.Comparethetotalcomputationaleffort

hesame calculationbytheuseofTheVenin'stheorem,

ethodisnotalwaysshorter.

bove,computetheTheVeninequivalentvoltagee,and

theleft-handcircuitis anequivalentonesofaras the

throughthe resistanceRisconcerned.Inthecomputa-

sformationisconvenient.

ob.18 asanexerciseinNorton'stheorem,andcarry

mtoobtainanexpressionforthe ratioet/iiasa func-

viousproblem,applythecurrentii attheright-hand

geatthe left-handend.Showthatthesameexpression

verifythereciprocitytheoreminthisinstance.


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shown,computetheopen-circuitdriving-pointand

emethoddiscussedforladdernetworks,andthusobtain

alteecircuit.Alternatelyobtainthesameresultthrough

s.Contrastthetotalcomputationsinvolvedinthetwo

tworkoftheprecedingproblemwitha 2-ohmresistance

dtheequivalenttee,pi, andsymmetrical-latticenetworks.

alpairshowninto (a)anequivalenttee,(b)an equiva-

ymmetricallattice.Computevaluesoftheopen-circuit

mpedances,theshort-circuitdriving-pointandtransfer

eralcircuitparameters.

ircuitparametersoftwocascadedtwoterminal-pair

lparametersareA\,B\,C\,D\andAt,B%Ci,D% Write

fthisresult when(a)thetwonetworksareidentical,

enticaland individuallysymmetrical.Usingthelatter

ametersfora cascadeoftenidenticalteesectionsas


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ANDRELATEDTOPICS

ydissymmetricalnetworkgivenhereisnevertheless

ndfinda phys-

lent.

callattice

ancedladder

1-ohmresist-

voltage

owerto aload

sistanceR.Ift, =10amperesand

mpute(a)thepower

rce,(b)the

agesource,(c)thepowerabsorbedby theload.For

ourcessupplyequalamountsofpower?

e situationinProb.28,andcarrythroughthe corre-

erecomputethenet powerabsorbedbyfirstfinding

branches.Nextcomputethetotalpowerabsorbedsepa-

esactingaloneandfor thecurrentsourcesactingalone.

evaluesequalsthetotalpowerfor allsourcesacting

fProb.30,computethe powersuppliedbythecurrent

thevoltagesourcessimultaneouslyacting,andnotethe

powerdueto aninsertionofthevoltagesources.

epowersuppliedbythevoltagesources,withandwith-

multaneouslyacting,andnotethedifferenceinthe supplied

nofthe currentsources.Whatcanyouconcludeasto

rduetovoltageand currentsources?


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ddernetworkofProb.30 withthevoltagesourcesand

rceremoved.Computetheratioofvoltageatthe right

hevalueofcurrentin thecentralmesh.

or whichtheimpedancelevelofthecentralmeshis

putethetransfervoltage-to-currentratioandthecurrent

utethepowerinputin bothcases.Whatconclusions

ults?

nsiderationsshowthattheschematicrepresentationof

ncedtransmissionlineswithassociatedsourcesandloads

ereplacedby theso-calledsingle-linediagraminsketch(b)

onofthevoltagesei,e^,ej,andthe currentsintheresistances

Thecommondatum(ground)isindicatedbyshading.

wsasingle-linediagramofthe sortdiscussedinthe

stancesare1ohm,andthevoltagesare fixedbysources

ghappropriatesourcemanipulationsdeterminedirectly

henodepotential e.Thenfindallbranchand source

epowersuppliedbyeachsource.


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onsoftheElements

whichalinearpassivenetworkis builtare

ysicalrealityin themselves.Thisfactbe-

ndedthatwhat,for purposesofanalysis,we

ediatelyidentifiablewithitsphysicalcounter-

uswhosebehaviorwewishtostudy.Several

pproximationneedusuallytobegone through

mthatwe

t.Toac-

ropriately

tpart

ne that

knowl-

tRinfd8eof networktheorybutskilland

mhos.judgmentaswell.Sufficeit tosay

stal-

schematicnetworkconfigurationwhich,for

esentsthebehaviorofthephysicalapparatus

es.Thisschematicrepresentationoftheactual

tricalnetwork.Itselementsexistonlyby

Letusreviewandelaborateuponthesedefi-

resistanceorconductance,isshownin

weenvoltagee(t)andcurrenti(t) atits

heequation

ndconductanceGhavereciprocalvalues.

nti(t) arearbitraryfunctionsofthetime,


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IONSOFTHEELEMENTS

dtoeachother byEq.1.SinceR orGis a

aresimplyproportionaltoeachother;thatis

edbycurvesof exactlythesameshapewhen

maybothbeaccommodatedbythesame

toits

efactors

Thus,

con-

(t)is a

ooking

efunny-

tmakes

isan

esult-

ayabout.

lation1.

ogetherwithitsvolt-amperecharacteristic

tLinhenrys

nreciprocal

and

existsnoestablishedsymbolornamefor

earbitrarytimefunctionsexceptforthe

em.Thisre-

proportion

ment,but

ntegration.

arpulse,as

constant

and linearly

eintervalas

figure.Except

stantLor

ctione(<)shown

t),and,con-

ofe(t),as

shapesof

thingelse,exceptthate(t)wouldalwayshave

erivativeofi(t)withLequal totheconstant

textisanupsidedown LasinEq. 2.

ofan


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SOURCEFUNCTIONS

relationshipremainsthesamewhethere(t)

eresponseorviceversa.

pacitance—isshowninFig.4.Herealso,

risticisexpressedthroughdifferentiationor

expressedeitherintermsofC infaradsorS

stanceelement,establishednamesandsym-

he ca-

cal,the

thecase

teristic

tanceele-

entCin,.., ..,,

arafs.mentareentire,ysUnilart0thoSe

ceele-

fferencethattheidentitiesof currentand

ationsexpressedinEqs. 1,2,and3 reveals

y,thatasagroup theyremainunalteredif

R withG,andLwith C(orL~1withS).

bemadeuse ofextensivelylateron.It

calledprincipleofdualitywhichcapitalizes

ral invariantpropertiesofnetworksmay

esimpleones revealedhere.

rces

rebuiltbyinterconnectinginanywaythe

Beforewehavea problemofanalysis

etworkmustinsomewaybe excited;thatis,

bepresent.Sincetheinductanceandcapaci-

leofstoringenergyintheir associatedmag-

ffectivewayofregardinga sourceincon-

udiesis toconsideritasa devicethatinserts

ntooneormoreofthestorageelementsL or

capableofenergystorage,playsnopartin

n).Asin theconsiderationofresistivenet-

tosupplyenergyis regardedasbeingoneof

agesourceoracurrentsource.Theseare

nFig.5.Botharefictitioussincetheyexist


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OURCES

thecaseofthenetworkelementsthey

ropriateelements,fortherepresentationof

cesatitsterminalsthe voltagee,(t)which

itisappropriateto considerinviewofthe

terminalvoltageis e,(t)bydefinition,it

anycircumstancesofenvironmentinwhich

Thatis tosay,theterminalvoltageremains

ofwhatmaybeplacedacrosstheseterminals

dacrossthe terminalsofthevoltagesource,

e,amperesisestablished,andthesource

joulesof energypersecond.AsGbecomes

eslargerandsodoes thepowerdelivered.If

rossthesource,thenaninfinitecurrentexists

red,butthe terminalvoltageremainsun-

nsibletoshort-circuitavoltagesource;itis

Nevertheless,ifweimaginetheterminals

nitecurrentexistinginthe closedloopformed

htheshort-circuitingpath,thenwerealize

be ashortcircuit,otherwisethefinitevoltage

op couldnotbeproducinginfinitecurrent.

:namely,thatthevoltagesourceitself isa

na networkinsertsavoltagee,(t)and

urceis bridgedacrosstwopointsina net-

voltagee,(t)existingbetweenthem,these

hevoltagesourceis toregarditas a

mentforwhichthevolt-ampererelationis

he samevalueregardlessofthecurrent,

especifythat e,(t)=0,thenthe voltage

ort circuit,forashortcircuitis adevice

zeroregardlessofthe current.Likeashort

urce

nopencircuit.

dlewhenshort-circuited

sourcesorconstraints.


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NDSOURCEFUNCTIONS

isaconstraint;butit ismoregeneralinthat

ts terminalstoanydesiredvalue.In

t avoltagesourceisa generalizedshort

mannerthecurrentsourceofFig. 5is

rovidesa statedcurrenti,(t)atits terminals.

si,(t) bydefinition,itcannotbealtered

esofenvironmentinwhichthesourcemay

heterminalcurrentremainspreciselyt,(0,

placedacrosstheseterminals.If aresistance

nalsof thecurrentsource,thenavoltage

shedandthesourcedeliversRi,2wattsor

cond.AsR becomeslarger,thevoltage

esthepowerdelivered.Onopencircuit,the

ninfinitevoltageanddeliversinfinitepower.

ircuitacurrentsource;itis idlewhenshort-

gardedasa peculiarkindofcircuitelement

onspecifiesthatthecurrentisindependentof

(t)= 0,thecurrentsourceisidentical

opencircuitisadevicethatforcesthe cur-

oftheterminalvoltage.Likeanopencircuit,

nstraint;butitismoregeneral inthatitcon-

erminalstoanydesiredvalue.Inthis sense

ntsourceisageneralizedopencircuit.

sedto deliverchargetoacapacitance,this

milartotheprocessof fillingatumblerwith

tterplayingthe roleof"source"andthe

pacitance.Thewaterisregardedasanalogous

s theanalogueofcurrent.Ifthevelocity

pidly.Justso,fora largesourcecurrent

capacitancetowhichitconnectsmounts

rinthenormalmannerrequiresa finite

ethatwecouldhavethewaterpackagedbefore-

mbler—"kerplunk."Inthiscasethetum-

bstantiallyso.Analogouslywemight

cedeliveringapackagedamountofcharge

citanceandraisingitsvoltagefromnothing

Sucha kindofsourcefunctioniscalledan

ful roleinourlater work,anditis worth

rtiesmorecarefully.


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OURCES

nctionthatdeliversa finiteamountof

t (a)ofFig.6.The chargeq(t),whichis

rrent,isshowninpart(b) ofthesamefigure.

5seconds.Thesourcecurrentisconstant

nterval,andzerootherwise.Thecharge

e intervalS,reachingthefinalvalue/, X6,

thecurrent

redtoas a

ralq(t)is

ulse isthe

dthelatter is

Graphically,

weenthetwo

lizable.

he pair

istruefor

weversmall.

,X 5,the

se,or thenet

tsayQ

medto be

seis talland

rom zerotothevalueQ atarapidrate.As

aller,thecurrentpulseapproachescloserand

elimit8—* 0thesourcecurrentis zero

0whereit isinfinite;neverthelessinthis

ulse(whichis thencalledanimpulse)isstill

sameareaQ.Thelinear rampfunctionq(t)

a stepfunction.

nity,thelimitingcurrentfunctioniscalled

itingchargefunctionaunitstep.Otherwise

ctivelyasan impulseofvalueQanda stepof

etonotethati,(t)remainsthederivative

alofi,(t)atany stageinthelimitingprocess

hatoneispermittedto applythisinterpre-

alrelationshipbetweeni,(t)andq(t) even

ativeofastepfunctionof valueQisanim-

ntegralofanimpulseof valueQisastep

asthusgaineda usefulandmathematically

ederivativeofafunctionat apointofdis-

ction

argeqit).


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SOURCEFUNCTIONS

tsinvolvedintheconceptsof"pulse

,"impulse,"and"step"havebeenpresented

ce,theymayequallywellbeappliedto avolt-

ttimesfind itappropriatetoconsidera

pulse,oraramp,or animpulse,orastep.

nshows,however,oneusuallyhasachoice

tobeconsideredasavoltageor asacurrent,

attheactualphysicalsituationtobe por-

onlythe situationofacurrentsourcecon-

ement.Suppose,instead,weconsiderthe

totheinductanceelementofFig.2,andin-

thevoltagedevelopedacrossthisinductance.

msofthepertinentvolt-ampererelation,

geis proportionaltothederivativeofthe

yconstantbeingthevalueLofthe induct-

rst,supposethe currenti,(t)feedingthe

mpfunctionlikeq(t)in Fig.6.Itis clearat

evelopedacrosstheinductancehasthe form

part (a)ofFig.6.If therampi,(t)hasa

eS,thenthe voltagepulsee(t)hasaheight

ematters areclearbyinspectionofthe

andthebasicnotionthatthe derivativeofa

pe.Theslope ofthelinearrampis anon-

seinterval,and zerootherwise.Notethat

ularpulsee(t) is(height)X(duration)

ndependentof5. Therefore,ifwenow

ess8—>0, thecurrentfunctioni,(t)ap-

e I,ande(<)approachesanimpulseofthe

esupposedthatthecurrentis theapplied

voltageisthe networkresponseor"effect."

mpererelationshipdeterminedbytheinduct-

dlessofwhichquantity,e ori,is the"cause"

Wearepermittedto conclude,therefore,

ulseappliedtoaninductanceelementpro-

tionresponse.Intheaboveconsiderations

evalueLI, andtheassociatedcurrentstep

mwherebythisflexibilityinviewpointis achieved,isthat

uitsinArt.5 ofCh.3,andwill begiveninmoregeneral


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OURCES

esevaluesaremultipliedbythe same

betweenvoltageandcurrentorcauseand

anoperationisalwayspermittedina linear

statethat,ifa unit-voltageimpulse(con-

pliedtoaninductanceL,theresponseis a

evalue1/Lamperes.

ngresultsthat wewishtocompare.A

edtoa capacitance,instantaneouslyplacesa

t capacitance;aunitvoltageimpulse applied

neouslycreatesafinitecurrentinthat induct-

weenthesetwostatementsmaybemade

efollowingphysicalconsiderations.Ifthe

acoil,thenthe voltageatitsterminalsis

sduetoa rateofchangeoffluxlinkages;

beingthenumberof turnsinthecoil andct>

definitionL=nct>/i,wenotethata current

dstoa fluxlinkagencj)ofunity.Thestate-

ceofthis paragraphmaynowbemademore

ulseappliedtoa capacitanceinstantaneously

omb)inthat capacitance;aunitvoltageim-

nceinstantaneouslyplacesunitfluxlinkage

ctance.

eaboutacurrentimpulseandthe other

reidenticalexceptforan interchangeof

ndi,Cand L,chargeandfluxlinkage.Or

statementismade,andthatthisone remains

hedualquantitiesin thepairsmentioned.

mpleofthe principleofdualitywhichwe

urdiscussionscontinue.

felectricchargeintoacapacitancerepre-

ofa finiteamountofenergytothesystemof

apart.QcoulombsinC faradsrepresentsan

hichmayalternativelybewrittenCEc2/2if

geproducedinthe capacitancebythe

ddenintroductionoffluxlinkageintoan

eadditionofenergytothenetworkofwhich

ct>weber-turnsinL henrysrepresentsan

es,whichmayalternativelybewrittenLIl2/2

urrentproducedintheinductancebythe

fromthepresenttopicbutnevertheless

totheeffectthatsomereadersmay notlike


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NDSOURCEFUNCTIONS

nkageproducingcurrent.Theymayfeel

yabout,thatcurrentproducesfluxlinkage.

ersofelectricityandmagnetismhavecon-

uationinthis wayforaslongas thesubject

ctuallymorereason baseduponphysical

"interpretationofsuchpurelymathematical

gneticfieldsmakesanysenseatall) toadhere

omagneticfieldproducesvoltageandcurrent

eis true.Forpurposesofanalysisit does

interpretthemathematicalrelationships.

ibleviewof suchthingsandbereadyto

on,whicheverisconsistentwiththetenorof

entsaboutcurrentorvoltageimpulsesand

etworkelementswemaysay:A unitcurrent

itanceofCfaradsestablishesinstantlya

serts1/2Cjoules ofenergy;aunitvoltage

ctanceofLhenrys establishesinstantlya

n(henceacurrentof 1/Lamperes)andin-

Functions;SomePhysical

tionsintroducedinthe previousarticle

usefulbecausemanyactualexcitationfunc-

terms ofthem.Inthisregard,the step

ostwidelyknownofthe two,forithasbeen

ntheliteratureoncircuittheory formany

ucedthroughthewritingsofOliverHeaviside

enineteenthcentury.

sefulnessof suchaconceptasthestep

mmonlyoccurringsituationpicturedinpart

assivenetwork(shownbythebox)isassumed

ywith theconstantvalueofEvolts through

usuallyis tostudythenetworkresponse

closureoftheswitch,withthe assumption

beforethistime.

nthenetworkresponselongafterthe

hentheexcitationfunctionisregardedas a

hevalueE.However,iftheinterestLies

e networkimmediatelyfollowingtheswitch

ynot appropriatetoregardtheexcitationas


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RITYFUNCTIONS197

continuityinthisfunctionoccurringatthe

outstandingcharacteristic.Thatisto say,

theexcitationfromthezerovalue priortothe

zeroconstantvalueafterwardthatcharac-

etworkresponsenearthistimeinstant.

stantvoltageEthroughclosureof switch5atthe

atelyberepresentedwithoutreferencetoaswitchingopera-

s thestepfunctionshowninFig.8.

whichplacesthe discontinuousnatureof

,isenhancedthroughredrawingthephysical

(b)ofFig.7 wherethenetworkisregarded

esourcefunctione,(t),anddescribingthis

owninFig.8wherethe jumpine,(t)occurring

ture.

ytodescribethise,(t)byshowinga picture

vean

suchafunc-

tjumpare

ercamethis

mbolthat

onbydefi-

ueEat 1

renceatU>=0

donewith-

lledthefunctiontheunitstep,andindicated

oll(t).

osedtheseideas,mathematicianswere

ecausefunctionspossessingdiscontinuities

rforrespectablemathematicianstoassociate

egardedasablemishor evenworse—asort

sthatabsolutelybarreditspossessorfrom

tyofgenteelfunctionswhosebehaviormade

ematicians.Thegameofmathematicswas

strictcodein thosedays.

ltraconservativeattitudehaswornoff,

oallwhoareprimarilyinterestedin theuse-

onof

.


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oolsandconcepts,andwithnolossin therigor

ustified.Thuswe notonlyaccepttoday

tep,but wehavegonefartherinthat a

leinterpretationisgivento thederivativeof

ulse.Oncethisessentialhurdlehasbeen

blishmentofasimpleinterpretationforthe

pointofdiscontinuity,thewayisopen

higherderivativesatthispoint.

byreferencetoFig.9. Hereparts(a)and

ederivativeof theunitdiscontinuityis

both approachtheunitimpulseasS —>0,while(b)

velytheintegralandthederivativeof thisfunction.

n(b)isclearlytheintegralof function(a)or

unction(b) foranyfinite5,howeversmall.

smaller,(a)approachestheunit impulseand

usly,weseethatfunction(c)isthederiva-

n(d)isthe integralof(c)foranyfinite 5,

,functions(a)and(d)bothencloseunitarea

eachotheras5 becomessmall.Thus,with

achestheunitstep,functions(a)and (d)

,andfunction(c)approacheswhatcanbe

veoftheunit impulse.Since,forverysmall

wooppositelydirectedverytall spikesthat

her,thefunctioninthelimit iscalledaunit

uretoacoupleused inmechanicstoindicate

at somepoint.

usedtocircumventthedifficultyimposed

tinuityis toreplaceitbya gradualriseof

eticallywethenregardtherise intervalas

Solongastheri seintervalisnotactually

ion astotherigor orappropriatenessofthese

whetherornottheyarestill rigorousinthe


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ITYFUNCTIONS

mattertous, forpracticallyspeakingwecan

fficientlysmallnonzero8, sinceinnature

nabsolutelyabruptchange,and theengineer

occur innature.

cansimilarlyinterpretthe derivativeof

unittriplet),andthatone cantheoretically

achfunctionsuccessivelyobtained.

tioncanbeundonethroughsuccessiveinte-

een,theintegral oftheunitimpulseis the

eunitrampwhichis simplyalinearrisewith

tionyieldsaparabolicr ise,etc.

bsequentdischargeof acapacitanceCintoa network

sentedthroughassumingtheexcitationto beacurrent

milyof functionsthatarerelatedoneto

tiationorintegration.Thisfamilyis spoken

nssince, inthelightof conservativemathe-

thersingular.Any oneofthesefunctionsis

t)in whichthesubscriptnis referredtoas

yfunction,andsuccessivefunctionsarerelated

asthecentralfunctionof thisfamilyand

thesingularityfunctionofzeroorder.The

heunit rampu_2(0,theunitdoubletWi(<),

stheold Heavisidenotation1(<).

einstantofoccurrenceis tacitlyassumed

rring att= t0iswrittenun(t — to).

tyfunctionsastypicalkinds ofexcitation

lly recognizedbytheprecedingdiscussion

ateactualphysicalsituationsthrough the

ularityfunctionsasappliedvoltagesor cur-

urtherto illustrate.Thuswemighten-

redin Fig.10(a).HerethecapacitanceC

tery voltageEbyclosingswitchSi foran


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NDSOURCEFUNCTIONS

peningitagain.If nowswitchS2isclosed,

gedintothepassivenetworkinsidethebox.

atedasshownin part(b)ofFig. 10where

volvesnoswitches,andthecapacitanceC

urceequaltoanimpulseof valueEC.At

ccurs,achargeofECcoulombsissuddenly

nce,afterwhichthedischargeconditionsare

e circuitarrangementofpart(a)following

wingswitchesanddescribingtheir sequence

talsoadditionaladvantageshavingtodo

nctionmaybe regardedasasumof rectangularele-

ulsesastheircommonwidthA<ischosento besufficiently

ngthedischargephenomenonaccruefrom

oninFig.10(a)by thefictitiousoneinpart

erfeatureisdiscussedinthe nextchapter.

esingularityfunctionsareveryspecial

trictedclassofactualproblemscanbe treated

ary,onereadilyappreciatesthatanarbitrary

intermsofappropriatelyselectedsingularity

eisessentiallyanarrowrectangularpulse,

otheredtolet Sinthe sketchesofFig.9

strates,itis possibletorepresentanytime

uccessionoftheseelementarynarrowrec-

priatelyvaryingaltitude.Sincethevalue

alsitsenclosedarea,wecanvarythe height

valueoftheimpulseit represents,having

rall rectangularpulses.Althoughweshall

ilsomewhatlaterin thesediscussions,itis

eneralwaytheimportantfactthatthrough

nsesonecanconstructtheresponseofany

ryapplieddisturbance.


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RITYFUNCTIONS

einadaptingsingularityfunctionstothe

altypesofexcitationmaybeappreciatedby

gefunctionshowninFig.12,whichhassev-

ughoutfiniteintervalsandisdiscontinuous

us,for t<0,e,(t) =0;fromt =0to

t=titot=^itequals—1;andfort>i2

wemay writethisfunctionas

h)+u-i(t- t2)(5)

0and2for t>0;the secondtermiszero

i;and

<'

po-2

unc-1

nc-

he-l

ak-^ ^timefuncticmequa,

sumofstepfunctionsgivenby

mayEq.5.

curve,

the unitstepmaylikewisebeused asabuild-

onofarbitraryfunctions.

ulseithas beenpointedoutthattheen-

easureofthe"value"ofthis function.Again

uationforanonzero5 sincefor5=0 itis

alizewhatwemeanbytheterm "enclosed

theterm"valueofa function"todesignate

thatis characteristicaboutitsamplitude.

f(t)=10sin tissaid tobeasinusoid of

imilarlythe functionf(t)=3u—i(<)isa

d3 istheamountofits discontinuity;that

htheamplitudeofthe functionjumps.

,itis manifestlynotpossibletobasethe

ncethelatterisinfinite.Nevertheless,one

mpulsesofvarioussizesbecausetherectan-

erivedthroughan appropriatelimitingproc-

mountsofarea.It isthisareathat appears

n whichtobasethedistinctionbetweenim-

y.Inthe caseofacurrentimpulseapplied

seenthatthevalue oftheimpulseequalsthe

njectedintothe capacitance.Clearly,one


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mpulsesofdifferentvalue,fortheireffect

roportionalto thisindexofintensity.

utthissituationis thatitappearstoget

nally.Thusthetimeintegralof currentis

a current-versus-timecurveischarge.Con-

currentimpulsedimensionallyisa charge,

s.Wethusexpressthevalueofa current

reaswewouldnaturallyexpectthevalueofa

essedinamperes.Similarly,thevalueof a

ed,notinvolts,but inweber-turnsbecause

ereisfortunatelynotoffundamentalcon-

hingwrongwithexpressingthevalueof a

bsinsteadofamperessolongas itisnot the

eofacurrentthatweare referringtoasbeing

mbs.Theterm"value"inconnectionwith

edon aratherspecialdefinitionwhichunfor-

ethedimensionsofthe physicalquantity

lto beawareofthissituationso asnotto

econsequencesindimensionalreasoning,it

hinkingtobeannoyedbyit sinceitcanbe

respects.

ortanceisthequestionof howanimpulse

phicalrepresentationofan excitationfunc-

ousvalueofanimpulseis eithernothingor

oadoptsomeconventionalmethodforits

ntationforthefunctiongiveninEq. 6.

meusedinthepresenttextis illustratedin

unction

<i) +Au0(t-t2)(6)

a verticalarrowwiththeinfinitysign(co)

area)is expressedbythenumberorsymbol

Anegativeimpulseis drawnasanarrow


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ATIONS

irection;oronecouldalternativelyindicate

negativealgebraicsigntotheassociated

owinFig. 13couldalternativelybedrawn

en—4.6.

urse,tolettheheightof thearrow,accord-

,indicatethevalueoftheimpulse.However,

se isdimensionallynotthesameasthe ordi-

onitrepresents,it seemsbettertodrawall

lengthand toindicatevaluesbyplacingthe

mbolsadjacenttothem.

herorderthantheimpulse areusedso

essentialtoformulateappropriategraphical

tthistime.

ations

detaileddiscussionwasgivenfornetworks

ementsalone.Analogouslyonemayencoun-

networksconsistingofcapacitanceelements

mentsalone.Wereferto networksofthis

es.Themethodofdeterminingthevoltage

roughoutsuchasingle-elementnetworkis

heritberesistive,capacitive,orinductivein

cussiongiveninthepreviouschapterswith

orksappliesinessentiallyunalteredformto

inductancenetworksaswell.Afewcom-

eristicdifferencesincertainspecificdetails,

workconsistingofcapacitancesaloneand

tiontothepertinentvolt-ampererelation3.

etweenthisvolt-ampererelationandtheone

eelement,asgivenby1, liesinthefactthat

njectstheoperationsofdifferentiationand

vethisdifferenceineitheroftwo ways.One

ofcurrent,theassociatedchargedefinedby

ieveasimilareffectbyusing,insteadofvolt-


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tivefunction

)

adofrelations3to characterizethecapaci-

resentingthepropertiesofthis elementin

eratherthanits

urrent)relation.

nt distribution

networkwefind

esubsequently

ather thanthe

,weneedmerelyto

atecharge(Eq.7).

stributionisin

knowledgeof

dtofind itwe

ons8 which,like

heresistanceele-

entiationorin-

canbeused to

ceelement.In

tributionofavoltage-derivativefunction

fvoltagethroughoutthe network.The

hen befoundbyintegratingthederiva-

dealinginthenetworkproblemwithrela-

tiveorintegral signs,andhencethispart of

arto thetreatmentofapurer esistance

enetworkof Fig.14.Herethenumbers

esservetoindicate thebranchnumbering

as indicatingthecapacitanceelementvalues

sumedtobeaunit currentimpulse,orex-

unit step.

henodebasis ischosenandthevoltage

ode-to-datumsetas indicated,theleft-hand

ngrelations8 wecanwriteforthe various


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ATIONS

equationsarewrittendirectlyintermsof

).AspointedoutinCh.2, Art.1,theseequa-

principleoftheconservationofchargeand

ormtothe chargesaswellasto thecurrents

echarges).WithreferencetoFig. 14itis

12 yieldsafterappropriatearrangement

)

rix

carefulexaminationoftheseresultsthat

qs.13couldhavebeen writtendownbyin-

gthepatternestablishedin Ch.2,Art.6for

etworksonthenodebasis. Thesituation

exceptthatwe dealwithcapacitancesinstead

getacapacitancematrix[C]insteadofa

andtheexcitationfunctionisachargeinstead


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SOURCEFUNCTIONS

ceedthroughregardingthe firsttwoof

elvesintheform

e aunitstep,it followsthatei,e2,€3are

eirvaluesbeingtheappropriatefractions

q,(t)weresomeothertime function,then

hertimefunctionexceptforthe scalefactors

essedby Eqs.17and18.

onecan readilycomputethevoltage

nce;andfromthisvoltageandthepertinent

ncomputetheassociatedcharge.Oronecan

rectlyfromrelations11.

betweenthedatumandnode3 isavolt-

sp. charge)thenthepotentiale3is thisvolt-

16thenyieldthesolutionimmediately.

n17fore3as afunctionofq„onehas the

node3anddatum;namely,thiscapacitance

thevoltagesinthevariousbranchesin

arepreciselythesameastheywouldbeif

placedbyaconductancewiththesamevalue

cevalueinfarads,andiftheexcitationwerea

analogousresistivenetworkonecomputes

osbetweenanytwonodes,theresult isthe

ceinfaradsbetweenthesametwo nodesin

Theanalogousresistancenetworkmayin

asortofreferencenetwork,sincethevoltage

rit isnumericallyidenticalwiththevoltage

theactualcapacitancenetwork.

sizeagainthatallof thevoltagesinthe

givenbythesourcechargeq,(t)multiplied


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ATIONS

oltagesastime functionsarepreciselythe

q,(t).In thisexampleq,(t)isaunit step

dtobea unitimpulse.Thus,whenacapaci-

tsomepointby acurrentimpulse,thevolt-

workarestepfunctions.If theexcitation

egral oftheimpulse),thenallvoltageswould

egralofthe step).Thecapacitancenetwork,

metrymaybe,hasforitsdriving-pointand

ionsthesamefundamentalmathematical

dforthe singlecapacitanceelementbyEqs.3.

ulto pointoutthattheunits impliedin

onofcapacitancevaluesneednotbeconsidered

utionto acapacitancenetworksuchasthe

valuesarespecifiedinthesameunits. For

licity,thegivennumericalvalueswereas-

meonemightsuggestthatwearebeingunreal-

2-, 3-,4-,and5-faradcapacitancesarelarger

ircuits.This circumstanceisrathertrivial,

nswehaveobtainedabovearereadilyadjusted

in capacitancevalues.Thusifwewereto

,3, 4,and5microfarads,allvoltagesfor the

argebecome106timeslarger;or allvoltages

plied currentequaltoone-millionthofaunit

ensibleto assumethecapacitancevalues

ourseofthesolution andinsertappropriate

ouldbeto writethefactor10-6aboutfifty

ssof numericalsolution,tosaynothingof

10-6X10-6=10-12etc.whencarryingout

weenpairsofequations.Beingrealistic

sisjustplainfoolishness.

s seenthattheproblemoffinding the

urrentthroughoutanetworkconsistingof

ntiallythesameasit isfora resistancenetwork.

sfactis touse,insteadofvoltage,theasso-

by


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evethedesiredresultthroughusing,instead

derivativefunction

econvertedtoa forminwhichdifferentiation

mely,

ntingthecharacteristicsoftheinductance

xlinkage-currentrelationinsteadofitsvolt-

it isavoltage-(cur-

therthantheconven-

tionthatisusedto

ceelement.Eithertrick

ndintegralsigns,and

nanalysisprocedure

othat usedwithresist-

stratedby meansof

rtheinductancenet-

ebranchnumbering

the elementvaluesin

medtobe aunitvolt-

geequaltoaunit step.

thatinFig.14 with

ductanceandcurrent

samekindofvoltage

eloopbasis ischosen,andthecurrent

et ofmeshcurrentsasindicated.Using

thevariousbranchfluxlinkages(those

nductances)


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NATIONS

equationswhichnormallyarewrittenin

reinsteadwrittendirectly intermsofthe

ages;thatis, intermsofthe branchflux

into24 yieldsafterappropriatearrange-

matrixthatcould havebeenwrittenby

nFig.15.

onsin set25forii andi2in termsofi3,

f Eqs.25thenyields

9)

aunitstep,all thecurrentsarestepfunc-

ageimpulseproducescurrentstep func-

inductancenetworkthesameasit would

,(t)wereanyothertimefunction,thenall

integralofthistimefunctionmultipliedby

29.

e valueofthenetinductancebetweenthe

ly,


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einductancesinthisnetworkwerereplaced

inohmsequalto thepertinentinductance

voltageexcitationwereaunitstepinstead

libriumequationswouldbeidenticalwith

theresultingcurrentvalueswouldbethe

ofthenetworkofFig.15 betweenanynode

eas thenetre-

enodepairinthis

work.

hangeinthe

tance,remarks

hecapacitance

aboveinduct-

rysinsteadof

esameapplied

slarger,orthey

pliedvoltageis

maller.

t,ifwehad

nthenetworkof

tanceswiththe

eciprocalhenrys,

ode basiswould

Eqs.13 forthe

g.14.Withrefer-

hesechangesare

esfa,fa,faarethetime integralsofthe

>that istosay,theyare nodefluxlinkages.

ntsasusualbyji,32, ••.we haveaccord-

ncurrentand fluxlinkageinaninduct-

Eq.20)

t-lawequationsread

rk

e.

on

ogous

g.


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SFORPARAMETERS211

Eqs. 32yieldsthedesirednodeequations

ancematrix

betweenEqs.11,12,13,and 14pertaining

rkofFig.14andEqs.31, 32,33,and34per-

ductancenetworkofFig.16.

of Fig.14,theelementvalueswerere-

rafsinsteadofcapacitancesinfarads,andif

orkwereexpressedonthe loopbasisassum-

tobea voltage,theresultswouldbenumer-

expressedbyEqs.23,24, 25,and26with

cenetworkofFig.15.

videnceoftheprinciple ofdualitypointed

precedingdiscussions.Weseethatthesame

epresenttheequilibriumofaresistance,or

cenetworkwithappropriateinterpretationof

tionterms.Theprincipalconclusionofvalue

ationatthemomentisthe factthatanyprob-

nationofvoltagesorcurrentsin acapacitance

tworkcanbefoundthroughconsideringthe

ppropriatelychosenresistancenetwork.Or

e-elementnetworkanalysisproblemisessen-

ofwhichof thethreekindsofcircuitelements

SFORPARAMETERSOF

NFIGURATION

udinallyuniformconductoroflengthI

by


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fthematerial.Forannealedcopperatroom

0-si nmksunits.

ngthofa (theoreticallyinfinitely)long

ofsolenoidlength(2)

tyoftheassumeduniformisotropicmedium

mersed,Aisthecross-sectionalarea(assumed

irediameter),andnequals theturnsper

At X10—7inmksunits. Then

ermeter(3)

rresultsunless thelengthofthesolenoid

sdiameter.

yshape itisbestto proceedfromthefol-

el oop,iscomputedbyintegratingaquantity

eticvectorpotential—aroundtheloop.This

onofdistances measuredalongtheloop,

pointp ontheloopbymeansof theintegral

path lengthdstata variablepointq,the

gentstothe loopatthepointsp andq,and

thesetwopoints.

eintegralsmayreadilybe evaluatedapprox-

solelyuponthegeometryinvolved.Weshall

esently,butfirstobservethatsubstitution

ousthattheinductanceparameter,asjust

metricalconstant,and(assumingy.tobea


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SFORPARAMETERS

hichitisinthe electromagneticsystemof

a length.Thustheunitof inductanceis

itsomecharacteristiclength.Thepractical

xedbychoosingitequal tothelengthofthe

0,000kilometers,or107meters,or 109

alizedmkssystemofunits inwhich

reads

thedoubleintegral equals107meters—the

ant.

alinEq. 6(thefirstone tobecomputed

8) isfacilitatedthroughreferencetoFig.1

oximateevaluationofthei ntegral6bythefinite sum9.

sareregardedasfinite,and thefixedincre-

istances5P9aremeasuredand towhichthe

chosenin thehorizontalpositionandlabeled

pproximatedbyafinitesumwhichyields

ormostpracticalcases.Notethattheangle

m zerotosubstantially360° sothat

aspositivevalues.

thusevaluatedapproximatelyby

formpathincrement.Thetermforq= 1

tlyrequiresspecialinterpretationastothe

of inductanceisthecentimeterwhichisequivalent


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Asjfrom itself,whichisnotzeroas might

revaluationneedsto considerthefinite

ngtheloop andinvolvesanintegration

ectionofthewire aswellasan integration

lement.Itturnsoutfrom suchanevaluation

1)

turallogarithmandris thewireradius(i.e.,

ror thickness).Theintegral9writtenfor

-»+T1A sc os6pt

Eq.8isobtainedthroughmultiplicationby

typicalAsp.Forthe firstterminEq.11

plicationbythe totalpathlengthI= nAs.

£

ioninvolvesalltermsforq =1,•• •,nexcept

hp =1,makingaltogethern(n— 1)

chFig.1isa regularpolygon(theapproxi-

l sumsoverqfor p=1,2, •••,n areiden-

m isthenequalto ntimesthesumfor asingle

oop wethushave

(13)

letuscomputethei nductanceofacircular

ximatingitgeometricallybyaninscribed

rfirstaninscribedsquareasshownin Fig.2

pposewe choosethethicknessofthewire

As/r=50\/2,andthefirst terminthe

value

)

umwhichisthe secondterminEq.13,two

nentcosinesarezero,and thethirdequals


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SFORPARAMETERS215

cos0i9=— 1.Henceformula13yields

7iX8.52henry(15)

erof thesquare.

edhexagonasshowninFig.3. Here

cir-

quare.

cir-

ex-

50as beforewenowhaveforthefirst

13

msinvolving

1/2


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NDSOURCEFUNCTIONS

thevalue15notonly intheratio8.74/8.52

eterof theinscribedhexagonislargerthan

.

circularl oopbyaninscribedoctagon.

hownin Fig.4wehave

eyields

.2311.7070.924

=0.765,theratioR/r= 50correspondsto

firsttermin thebracketofEq.13

uctanceinthiscase

10~7lX8.77

sthisresult isnearenoughtothe induct-

nfactmanycaseswill permitaneven

thisregarditshould benoticedinall three

tthesum involvedinEq.13contributesa

thevalueofthe loopinductance,sothata


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SFORPARAMETERS217

he approximateformula

hemys(20)

crohenryspermeter(21)

omenumericalvalues.Sincethelogarithm

r

inductanceisnot excitinglydependentupon

almostsaythatanyoldfair-sizedwire has

ancepermeter,andthe shapehasroughly

xtremecasesmust,ofcourse, beexcluded;

riginof theapproximateformula21(asgiven

),forit servesasavaluableguidein itsuse.

ofparallelplateshavingaseparationd is

earea (22)

oftheassumeduniformisotropicmediumin

rsed.Likethe solenoidformulaforLthis

platesareinfiniteintheir dimensions.Fora

mulayieldsonlyapproximateresults,butthese

estdimensionenteringintoAis stilllarge

ulfilledin practicalcases).Forfreespace

ts,and soonefinds

din meters.)Forapairofconcentric spheres

nthis caseexactly)

Ri=d,whichchecks theparallelplate


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NDSOURCEFUNCTIONS

antcaseR2—* &i— onehas

microfarads(25)

gsketchesofafunctionf(x),(a)drawthecurve

edf(x)/dx.

crohenrysisto beconstructedoutofcoppertubing

meterwoundintothe shapeofahelixwith apitchofabout

terof30 centimeters.Approximatelyhowlongapieceof

ow longwillthehelixbe?

tubingis2millimeters,computetheapproximate

etermineits timeconstantinseconds.

swoundon a5-centimeter-diametercylindricalcore

eofabout1millimeterdiametercloselyspacedfromturn

oximateinductance,resistance,andtimeconstantofthis

meters.

s woundoutofenameledcopperwireofabout1 milli-

eturns packedtogethersothattheresultantshapeis

andiameterof thedoughnutis20centimetersandthe

mputetheapproximateinductance,resistance,andtime

fcommunicationapparatusiswiredwithenameled

eterofabout0.5 millimeter.Incompletingthejob,

re usedaltogether.Estimatetheorderofmagnitudeof

nsertedintothecircuitrybythelead wires.

econstructedofanumberof dovetailedparallelplates

commonseparationof1millimeter.Abouthowmany

btainacapacitanceof0.005microfarad?Ifeachplatehas

s,whatwill betheapproximateoutsidedimensionsofthis


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ofsomepieceofelectricalapparatushasa diameterof

pproximatelyinthecenterofacubicallyshapedroom,

redwitha groundednetting.Ifthecommondimension

computetheapproximatecapacitanceoftheelectrode.If

apotentialof5X 10* voltsaboveground,computethe

eenergystoredin joules.Ifthisenergycouldbe released

tbulb atnormalincandescence,howlongwouldthebulb

rallyconsistsofaparallelcombinationofIn, Li,and

Listobe 5(allvaluesin microhenrys),whatmust

4,5 microfaradsrespectivelyareconnectedinseries

putethepotentialdrop acrosseachcapacitance,the

rge,thenet energystored,andthedistributionofthis

iouscapacitances.Supposethegivencapacitancevalues

farads,whatchangesdoyoumakein theaboveanswers?

4, 5millihenrysrespectivelyareconnectedinparallel

Computethecurrenttakenbyeachinductance,theflux

x linkage,thenetstoredenergy,anditsdistribution

uctances.Supposethegiveninductancevaluesarehenrys

atchangesdoyoumakeinthe aboveanswers?

aunit currentimpulseapplied.

raunit voltageimpulseapplied.

oltageimpulseappliedtoaninductanceLinstantly

amperes,whatvalueof currentisestablishedbythis

consistingofR andLinseries?Again,for acircuitcon-

es?

rrentimpulseappliedtoacapacitanceCinstantly

/Cvolts,whatvalueof potentialisestablishedbythis

consistingof RandCin parallel?Again,foracircuit

parallel?

appliedtoa circuitconsistingofR,L,andCin series,

ofar astheresultantcurrentis concerned,mayberegarded

chargingthe

hereafter

ircuitedupon

rnaturalbe-

hargeofthe

appliedto

and Cinpar-

fect,sofaras

cerned,may

tosuddenly

nceacurrent

rleavingthe

tstoperformwhatevernaturalbehaviorresultsfromthe

ce.

ketchthecapacitancevaluesareinmicrofarads.Com-

betweentheterminalsoandb. Repeatfortheterminal


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SOURCEFUNCTIONS

of1voltia appliedtotheterminalsc-d,computethe

hcapacitanceandthechargeoneach.Computetheenergy

andthroughtheadditionof theseresultsgetthetotal

attervaluethroughcomputingthenetenergyfromthe

candd.Iftheappliedvoltageis raisedto316,howdothe

Howdothevaluesof energychange?

henetworkof Prob.17arereplacedbyinductances

sarethesame,and ifasteadydirectcurrentof 1ampere

pairc-d,whatarethecurrentsin thevariousbranches?

esin thevariousinductances,andwhatisthetotalstored

wdothese valueschangeiftheappliedcurrentis changed

appliedtothe terminalpairc-dinthe networkof

tingterminalcurrentinnatureandvalue?Ditto forall

evariousbranches.

seisappliedtothe terminalpairc-dofProb.17,what

natureandvalueforall thevariousbranchesaswellas

isthetotalenergysupplied,and howisitdistributed?

urationofProb.17with capacitancesreplacedbyin-

nProb.18,a unitvoltageimpulseisappliedto theterminal

tureandvalueof theresultingcurrentineachbranch

ntcurrent.Computetheenergystoredineachbranchand

fR= 1ohm,L=10-3 henry,C— 10-4faradhas

pofthevalue1000.Forthe initialinstant,computevalues

nductance,itsrate ofincrease,thevoltageacrossthe

ndsecondderivatives.

tionofProb.17, allthecapacitancesarereplacedby

ttheoneacrossterminalsa-cwhichis2 ohms,andtheone

-bwhichisreplacedbyan inductanceof1millihenry.A

pliedtotheterminalpairc-d.Computethe initialrate

ghtheinductance.

ove,the voltagesourceasafunctionoftimeis sketched

ot thecurrenti(0,showingseparatelythecomponent

acitanceandinthe4-henryinductance.

0shownbelowisi mpresseduponacapacitanceC.

oftheimpulsesothat thevoltageacrossCbecomes


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nctionof Prob.25isinvolvedin theaccompanying

e(<).

rcesandtheresults theyproduceinasinglecircuit

ofelementanditsvalue inhenrys,farads,orohms,asthe

ResultantCurrentorVoltage

currenti,(<)=sin ttotheterminals ofaboxcontain-

sivenetworkproducesasteadyvoltageat theterminals

+30°). Whatcanyoudeduceastothe contentsofthebox?

s questionchangeife(l)— 10sin(<— 30°)?


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nResponse

neralPropertiesoftheSolution

mbinationofresistanceandinductance,

esourcee,(t).Theresulting currentis

Kirchhoff'svoltagelawandthevolt-ampere

mentsasdiscussedinCh.4,we havethe

edfori=i {t)withagivene,(t).Before

proceedingwith theformulationofasolution,

tandlookcare-

quationwehavebe-

ecandiscoversome

ur solutionwillhave

to findone.Some

ly thingto

-contemplatedoing;infacttheymayeven

o sofarasto exclaim:"Howcanyou

pertiesofa

und?"Weshallsee.

oursolutionwill resultasaconsequence

atis,the factthate,(<),i(0iandits derivative

ypowerotherthanunity,orthat thereare

ducte, andi.Becauseofthis propertyofthe

ansayatonce that,ife,(t)werereplacedby

stant,theni(t)becomesreplacedbyAi(t).

thisstatementthroughconsideringeach

thesameconstantA asin


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RALPROPERTIESOFSOLUTION223

e',(t)=Ae,(l)bea newvaluefortheex-

comesthecorrespondingnewvaluefor the

atthisconclusionwouldnotbepermissibleif

enteredwiththesquarepoweror anypower

equationcontainedtermsinvolvingthe

justmentionedweseethat inchoosing

t)weshallbefree toconsideranyscalefactor

e,sincethe solutionforanyotherscalefactor

multiplicationbythisfactor.Specifically,once

ya stepfunctionof1volt,we knowthatthe

of 673.596voltsissimply673.596times

erefore,neverneedtobotherin ouranalysis

rofvolts.Nomatterwhat theactualvoltage

lwaysgettheresponsepervoltfirst andthen

eafterward.Wecallthis processnormaliza-

on.Weshallsee thatsuchanormalization

erwhatour networklookslike,solongas

ndCare constant,forthentheequilibrium

ear.

owsbecauseofthelinearityof theequation

ei(0,wehavefoundasolutionii(JL),and, for

* wehavefoundthesolutiontobe givenby

,if weweretoconsidertheexcitatione(<)

nwouldbegivenbyi(t) =ii(t)+i2(f).

esuperposable,oradditive.Insomephysical

teranexcitationfunctionthatconsistsof

ei(0,e2(t),etc.We canthenobtainthecom-

deringeachpartof theexcitationasdeter-

m.Thecorrespondingseparatesolutionsmay

thertoform thecompletesolution.Thetruth

ilyverifiedthroughdirectsubstitutionin the

ervingthatitslinearcharacteris thefeature

wayofdealing withanotherwisecomplex

rimportantpropertyofoursolutionjust

sonealsofollowsbecauseofthe linearity,

atterhowcomplexthenetworkmayotherwise

erentiateeachterminEq.1withrespectto


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UNCTIONRESPONSE

exceptthat e,(t)andappearprimed.

sthat therelationbetweene',(t)andi'(t)

orkofFig.1 ispreciselythesameasis the

(t).Toillustratewith aspecificexample,

considerede,(<)tobea unitstep.Theni(t),

dit,would bethecorrespondingstep-function

esRLnetwork.Nowe',(<),beingthe deriva-

geimpulse,and,sincei'(t)is theresponse

atione',(t),wehavetheimportantresultthat

rnetworkcanbe hadbyfindingthestep-

erentiatingit.

differentiation,andviceversa,wemay

tep-functionresponsecouldbefoundthrough

responseandthen integratingtheresult.

presswhatwehavefoundoutabout therela-

i(t)bysaying thatoncewehavefoundi(t)

gardthesetwofunctionsasapair (asmates

elationshiphasonceand forallbeenfixed

e networkwhich,afterall,istheonlything

Whathappenstoone functioninthispah.

f oneisdifferentiated,theotheronebecomes

grated,theotherbecomesintegrated;if

twiceonone,itis repeatedtwiceontheother,

wayoflookingattheproblem,thatit is

ofunctions,e,(t)ori(t),we regardasexcita-

e.Thisstate ofaffairswaspointedoutpre-

sideringthevolt-ampererelationshipsforthe

.In Fig.1wehaveacombinationofR andL

ngmorethanthe volt-ampererelationatthe

mbination.Thevolt-ampererelationex-

t;andthisfacthasnothingto dowithwhether

chingfori(t) orwhetherwehappentoknow

e,(t).Letusnowdiscussthesolutionof Eq.1

.


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,(<)isa unitimpulse.Wefoundpreviously

,appliedto aninductanceLalone,instantly

nceacurrentof1/Lamperes.A bitofreflec-

onis notintheleast alteredbythepresence

usethefinitecurrent of1/Lamperesproduces

nitevoltagedropofR/Lvolts.Compared

t) attheinstantat whichthisimpulsehap-

tt= 0),anyfinitevoltagedropis negligible.

ent,whichthevolt-

blishesinthecircuit

esameforR andL

,(t)iszero forall

allyjustashortcircuit

t«=0,it suddenlyFiQ2SeriesRLcir-

peres.Thereforeourciutunder force-free

ththatoffindinghowcondition,

sedcircuitofFig.2

gthatatt= 0ithas thevalue1/L.The

nis expressedbythehomogeneousequation

ollowingtimefunction

uation,since

6 gives

edfor anonzeroi(t)throughsetting

tionexpressedby Eq.7readsmorespecifically

ntegrationwhichisdeterminedfromthe


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TIONRESPONSE

t=0. Writingthiscondition,oneobtains

rcuitofFig.1 thatresultswhene,(<)isa

rmofasimpledecayingexponentialasshown

albehaviorof theseriesRLcircuit.

hichdeterminesthe rateofdecayis

ant.ItsreciprocalL/Rhasthedimensionof

constant.Itisthatlength oftimeinwhich

0.368ofitsinitial value.Itisalsothat time

reachthevalue zeroifitcontinuedto de-

rationor differentiationoftimefunctions.

sisshownbythe factthatthetangenttothe

ezeroaxisat t=L/R.At/ =2L/R,that

ualtotwicethetime constant,thecurrenthas

ofits initialvalue,etc.Thetimeconstantis


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ewhenplottinga curvesuchasthatshownin

eresponseofthecircuitof Fig.1fore,(t)

vebuttointegratethe presentresultasgiven

re thatthedetailsofsucha processof

erstood,letusreviewafew typicalexamples,

essentialfeaturesinvolved.InFig.4the

yduringthe intervals\<t <2and

constantvalues1and —1respectively.

s theintegralofthis function;morespecifically

werefertosimplyas theintegraloffi(t),may

ulatedareaunderthecurve/i(<) from—«

upperlimitinthe integration.Asthisupper

tedareavaries.Itis thisvariationofthe

ththetimetthat thefunction/2(<)issup-

accordingtotheseideas,isshownbythe

2(t)mustbezeroforall timeuntilthebegin-

arpulseinoccurringatt= J.Atthis

owatunit ratebecause/i(<)=1.Thesame

roughouttheinterval\< t<2 andceases

eaccumulatedarea(valueof/2)equals1.5.

unchangedfromt =2untilt =4sincefi(t)

annotcauseacontributiontothenetarea.At

ginstodecreaseatunit rateandcontinues

chtimethenetaccumulatedareaequals— \.

)remainsat thisfinalvalue—\.

erdevelopafacilityforsketchinggraphically

phicallygivencurve,suchassketching/2(<)

n.Withthesamefacilityheshouldbe able

eprocess,recognizingatoncethat thepulse

veof thetrapezoidalfunctionfa{t).

elatedinthis mannerareshowninFig.5.

lsewitha discontinuityatitsleadingedge.

mulatedareaunderfi(t),exhibitsaparabolic

m t=0.5to t=1.5.Thereader should

dcharacteristicsinvolvedinthisexample,

tialrateofrise in/2(0equals6,whereforethe


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UNCTIONRESPONSE

eachesthevalue6 att=1.5 butthatthe

halfasgreat;further,thatthe risingportion

bolawith itsapexatthepoint wherethis

aximumvalue.Heshouldobservethat/2(<)

tiveatt= 0.5equaltothediscontinuityof

heparabolicrisein /2(<)aheadofthepoint

theconstantvaluebeyond;thatis,thereis

vativeof/2(<)att =1.5,asisclear fromthe

erivativeof/2) iscontinuousatt= 1.5.

eintegraloffi.

deaofwhatthestep-functionresponseof

klikebyconcentratingourattentionupon

veofFig.3andvisualizingthe processofintegra-

gralofa functionisgivengraphicallybythe

ingplottedcurve.Sincei(t)in Fig.3iszero

eintegralwillbezero forthisintervalalso.

curvesuddenlybeginsto growwithtime,

L.After t=0,the rateofgrowthofarea

nateofthe curveofFig.3forthat t,since

ate ofgrowthofitsintegral,accordingtothe

"derivative"and"integral."

wecansay—lookingcarefullyatFig.3—

mzeroatt= 0,increasesinitiallyattherate

teofgrowthpreciselyasthe curveofFig.3

heventuallybecomingzero.Thustheintegral

urvethat risesfromzeroandmonotonically

tewhichequalsthe totalareaunderthefunc-

t=».


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goodideaofwhattheanswershouldlook

analytically.Theformalintegrationyields

16)

dfromthefactthat thisexpressionmustbe

functionshowninFig.3.

tabove.Thuswe find

ecircuitofFig.1to anappliedunitvoltage

inFig.6and bearsoutwhatwasanticipated

elygraphicalreasoning.

.6is approachedexponentially,thecurrent

svalueatanyfinitetime. Inpracticewefind

ntsareusuallyverysmall—averysmallfrac-

tremecasesnot morethan1or2 seconds.

posesthecurrent,ina circuitofthissort,

einarather shorttime,afterwhichthevalue

dby theresistanceparameterRalone.

eininfluencingthenatureofthe response

up"periodextendingessentiallyoveran

o orthreetimeconstants.Thisintervalis

he"transientinterval"oralso asthe"interval

symptoticbehavioriscalledthe"steady


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TIONRESPONSE

"or"permanentstate."Inmanypractical

onlyin thepermanentstate,inwhichcasea

tionispossible,as willamplybediscussed

thematicalandPhysicalAspects

briumEq.1for thesimpleseriesRLcircuit

processofsolutionfromasomewhatdifferent

l becauseofthephysicalinterpretationsthat

tionsthesebeartothe purelymathematical

athereforconvenience,

thelineardifferentialequationwithconstant

(alsocalledanintegral)ofthisequationis

ucestheequationtothe identitye,(t)=e,(t).

ermsLidi/dt)+ Riwiththistimefunction

vertedintothe functione,(t).Itisreasonable

ctioni(t) thataccomplishesthisresultdepends

willdifferfordifferente,functions.Itis

icularfunctioni(t)thatsuitsthenature of

iscalledtheparticularintegralof Eq.19.

nctionbyip(t).

nconstitutesasolutioninthe sensethatit

n,wecanshowthatit isonlypartof amore

chlikewisesatisfiesthesameequation.This

nsider,inadditiontoEq.19,the one

mberiszero.A solutiontothisequationis

thatmakestheleft-handtermsaddto zero

eit isaparticularintegralfor e,(t)=0.

mefunctionbyio(t).Then itiseasilyappre-

makesthe termsL(di/dt)+Ri addto

sthe samease,(t).

tisfiesEq.20is calledthecomplementary

rticularintegralandcomplementaryfunc-

edtoas thecompletesolutiontothedifferential


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MATICALANDPHYSICALASPECTS231

ralreflectsonlythe natureofe,(t);the

ontheotherhand,containsthe appropriate

ughwhichthecompletesolutionachievesthe

arbitraryconditionsthatmaybespecified

allychosenas t=0).The followingdiscus-

emand givesaphysicalinterpretationtothe

spresentedabove.

veryoftenrepresentssomesteadyexcita-

ntvalueofa batteryord-cgenerator,orthe

ofasinusoidallytime-varyingvoltagewith

ever,asindi-

yexcitationmay

nectedtothe

switchSmay

aclosedpo-

whichthe

plated.The

heother7- Pertinenttothephys-

faswitch-

tionsolelythejngtransient

therefore

tthenetworkresponsewouldbeiftheswitch

meinthe pastandremainsclosedduringthe

Orwemaysaythatip(t)representsthe

rcuitata sufficientlylongtimeafteraninitial

atternof thisbehaviorissubstantiallycon-

(t)alone.Accordingtowhatispointedout

the precedingarticle,theparticularintegral

whatwerefertophysicallyasthe steady-

partofthe solutionreflectsthenatureofthe

wealsospeakofitas representingtheforced

s logicallysensiblethattheultimatebe-

thesuddenapplicationof adrivingforce

cterofthatforcefunctionalone.

ensibletoexpectthatthesteadypatternof

tnecessarilyestablishedinstantlyafterthe

othe network.Thusonemaythinkofthe

gthedictatorialdemandsofthe applied

siveandthereforehavingnosourceofenergy

countermeasures,itisgraduallybeateninto

eintensityanddurationoftheinitialstruggle

thedegreetowhichthebehaviorpattern

cediffersfromthat whichischaracteristic


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TIONRESPONSE

tter,whichwecall thenaturalbehavior

ssionof whatthenetworkchoosestodowhen

ted,orforce-free,statethenetworkbehavior

soweseethat thecomplementaryfunction

lythenaturalbehaviorofthenetwork.

ergesasasuperpositionofthe forcedand

orpatternsofthenetwork.Theforcedbehavior

orrespondsmathematicallytotheparticular

haviorornaturalresponseisidentifiedwiththe

Theirsumyieldsthe resultantbehavioror

forobtaininga solutionaccordingtothese

posethatthesourcee,(t) inFig.7is abattery

olts.If theinstantofswitchclosureis

differentialEq.19for t>0reads

ttheparticularintegralorsteady-statevalue

22gives

ecircuitis determinedbytheequation

fferentialequationisgivenbyan exponential

tants.Since

8)

elds

t±0),itfollowsthat


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MATICALANDPHYSICALASPECTS233

tion27maybewrittenmoreexplicitlyas

venbythe sumofEqs.25and 32,thus:

Ae~RtlL(33)

onentialpartof thesolutionplaysthepart

Itprovidesthesolution withthenecessary

aryinitial condition.Thusthennrrent™

ntf=0 mjjyxiit1-rn"g^ao™emjysjslahorate

beleftwithanonzerovaluei(0).Equation

orA intoEq.33thus givesthefinalresult

tlL(36)

tchecksEq.18 representingtheunitstep

uit.Thesecondterminvolvingi(0) showsthe

sturbanceresponsibleforthenonzerovalue

seffectevidentlyproceedsindependently

gto thesuddenapplicationofthevoltageE;

ethenetresponse.Ifthe circuitisinitially

condtermin Eq.36dropsout.

behaviorofthe circuitinrelationtothe

beseenfromEq. 32fori0(t)and Eq.35forthe

ThustheamplitudeAof thenaturalbe-

cebetweenthecurrentavailableinthecircuit

dbythe excitation.Thelargerthisdis-

be theamplitudeofi0(0-Thelatterfunction

gther oleofabufferor shockabsorber,since

discrepancymayexistbetweentheavailable

dbythesteady-stateresponse,andgradually

cysothatthenetresponsesmoothlyapproaches


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TIONRESPONSE

decaysexponentially,itseffectuponthe

onlyduringa relativelyshortintervalfollow-

n.Forthisreasonit isalsocalledthetransient

on.Itcomesinto existencethroughthesud-

drivingforceat theswitchinginstant,and

aritybetweenthesedemandsandthe available

nitiatinga transient,thecircuitisasserting

enchangeofstatewhichtheapplicationofan

uponit. Inresistingthechange,it effects

ew orderofthings.

oinitialdiscrepancyexists,orifthis dis-

ughsomeothermeanssuchastheapplication

onitis toestablishtheappropriateinitial

phyregardingtransientresponseoflinear

esregardlessoftheir degreesofcomplexity;

cedureappliesalsoinappropriatelyextended

ationlinkingcurrenti(t)insome partofa

ne,(t)atthesamepoint oranywhereelseis

^*

+boe'(37)

m• ..b0are realconstants(thea'sarealways

tnecessarilyso).

impleexampleofthistype.Thecomplete

mof aparticularintegralip(t)reflectingthe

ndacomplementaryfunctioni0(t)expressing

thenetwork.Thelatteris describedbyEq.

onto thisequationmayalwaysbefound

onentialform

39)

Eq.37yields,fore,= 0,

Oip+ Oo)i0(<)=0(40)


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N'STHEOREMS

hetrivialsolutioni0= 0,sowemusthave

ationpossessesnroots(p values)whichwe

nyone ofthesesubstitutedforP in

function;hencethemostgenerali0function

aviorthatis characteristicofthenetwork

ural tendencies),Eq.41isreferredtoas the

ndthePi,P2,''",Vn asthecharacteristicvalues

ethe termcomplexnaturalfrequencies)per-

iscussedandillustratedingreaterdetail

omentmerelyto callattentiontothefact

alwayshastheformip(t) +i0{t)inwhich

[for example,ife,(t)isaconstant,ip{t) isa

oid,ip(t)isa sinusoid],and-thetransientpart

venin Eq.42.ThequantitiesAi• ••Anare

irvaluesdependuponthediscrepanciesexist-

chargesincapacitancesandcurrentsinin-

ychosenastheswitchinginstant)andthose

statefunctionsforthesequantities.Ifjust

and currentsaresomehowpresentatt= 0,

ereis notransient.Inanycase,thetransient

eactiontothedemandsofthesuddenlyapplied

fferin effectingasmoothtransitionfromthe

y-stateresponse.

;Thevenin'sandNorton'sTheorems

may readilybeextendedtoanumberof

entRLcircuitwiththe helpofThevenin's

whichwe shallhereestablishina form

esentneeds.Withreferenceto part(a)of

edtocontainaperfectlyarbitrarylinear

geandcurrentsources.Thepairofterminals

tat random.BecausetheboxAcontains

oq=0

Ane*"'


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UNCTIONRESPONSE

elabeledeocwillbefoundto appearatthese

samebox Ahasconnectedacrossthe

theswitchS asecondboxBwhichalsocon-

workbutno sources.Anexternalsource

thboxesasshown.WiththeswitchS closed,

minals1-2isnotthe sameastheopen-circuit

rt(a)of thefigureunlessthecurrenti(t) is

a particularchoiceofthevoltagee,(t).Such

aysbebroughtabout;in factitisrather

ezeroif e,(t)=eoc(t)forthenthe voltage

ussionof Thevenin'stheorem.

tis open,iseoc(t)— e,(t)=0so thati(t)=0

switch<Sis openorclosed;nocurrententers

ver,bethoughtofas thealgebraicsumof

wouldexistfor e,(t)=0,andthe onethat

cesintheboxA becamezerobute,(t)were

ofthelinearityof allnetworkequations,such

ansimplybe addedtogethertogivethecor-

fallsources[includinge,(t)]actingsimul-

tioned.Sincefore,(t)— eM(t)thenetcurrent

osealgebraicsumequals thisnetcurrent

s inalgebraicsign)only.Therefore,the

= 0mayalternativelybecalculatedbyset-

Aequaltozero andinsertingavoltagee,(t)

othboxes.

ntsa rathergeneralsourcetransformation

vebeenthe embodimentofanelaborate

als1-2to whichanexternalsource-free(also

—theboxB—isattached.Whatwehavejust

at,sofar asthecurrententeringboxB is

arasthecompletebehaviorofthe networkin

ayconsidertheexcitationtobe lumpedat

boxes.


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N'STHEOREMS

— eoc(t)isto beregardedasafictitious

ssourceswithinthe boxAinitiallyrespon-

thevoltageeoc(l),whichis referredtoasan

encethesubscriptsoc)becauseitisthevoltage

terminals1-2whenthese arefree.When

serted,theoriginalsourcesin theboxAare

d itshouldberecalledatthis timethatadead

withashort circuitwhileadeadcurrentsource

cuit.Theselatter considerationsareim-

eavitalbearinguponthe geometricalstructure

ussionof Norton'stheorem.

workinbox Athatremainsafterthesources

o (aswillbeseenfrom exampleslateron).

equivalentexternalcurrentsourceinstead

cethe activecharacteroftheboxA.Figure9

turesofthistype ofsourceconversion.Here

1-2ofthe originalboxAbyitself [part(a)of

cuited,yieldingthecurrenti,c(t).Whenbox

nals1-2, asshowninpart(b) ofthesame

switchSbeingopen)a currentsourcei,(t)is

nalsalso.Thecurrentthroughtheterminals

esameas theshort-circuitcurrenti,c(t)shown

essthevoltagee(t) iscausedtobezero

eofthecurrenti,(t).Such aspecialcondition

out;infactit isratherobviousthate(t) will

for thenthecurrentthroughtheswitchS,

— i,(t)=0so thate(t)=0 regardlessof

sedor open;novoltageappearsacrossthe

ever,bethoughtofas thealgebraicsumof

wouldexistfor i,(t)=0,andthe onethat

cesintheboxA becamezerobuti,(t)were

ofthelinearityof allnetworkequations,such

cansimplybe addedtogethertogivethecor-

fallsources[includingi,(t)]actingsimul-


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TIONRESPONSE

,c(t)thenet voltageiszero,thetwovolt-

equalsthisnetvoltagecan differindirection

only.Therefore,thevoltagethatexistsfor

be calculatedthroughsettingallthesources

nsertingacurrenti,(l) =— i,c(0externally

zedinFigs.10 and11whichshowthealter-

vecharacterofa networkmaybereplaced

esultofTheVenin'stheorem.

.In Fig.10thisexternalsourceis avoltage;

eshouldclearlyobservethatthe equiva-

o whathappenstoasecondpassivecircuit

eterminals1-2and nottowhathappensin-

antvoltagesandcurrentsinsidetheboxA

efor thesingleexternalsourceastheyare

rces(althoughonecan determineinany

er).ThereplacementindicatedineitherFig.

esultofNorton'stheorem.

allysensibleonlywhentheinterestlies inthe

altotheboxA.

dinFig.10iscommonlyknownasThivenin's

hichis itsdual,asNorton'stheorem.

eequivalenceofvoltageandcurrent

eideasisshownin Fig.12,whereavoltage

resistanceRispresentedasbeingthe equiva-

inparallelwiththe sameresistanceRif

hisstatementbeingevidentuponrecognition

sameterminalvoltageon opencircuitand

onshortcircuit,and otherwisepresentthe

weentheseterminals(againtheshort-circuit


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N'STHEOREMS

ourceandtheopen-circuitcharacterofthe

ant).

ationinvolvingaresistanceelement.

ofthissortof equivalence,involvingasingle

wninFigs.13and 14.InFig.13 thepassive

L.Heretheequivalencedemandsthat(except

L)the voltagebethederivativeofthecurrent

ationinvolvinganinductanceelement.

ftheformer.Thus,forexample,ife,(t) in

heni,(t)isa stepfunction.

mentisacapacitance.Againthecurrent

elatedbydifferentiationorintegrationbut

ationinvolvingacapacitanceelement.

eto currentandvoltagereversed.Thatis,

e,(t)isa stepfunction.

upposewereturntotheconsiderationof

andapplytothevoltagesourcein serieswith

nshipgiveninFig.12.We thenobtainthe


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TIONRESPONSE

hat thecurrenti(t)throughthe inductance

elythesameas inFig.1if i,(t)issetequalto

e,ifi.(t)is (1/R)thofaunitimpulse,then

ce,ifi,(t)is a

timestheex-

nFig.15is

t(0isgiven

t step,theni(t)

ionin Eq.18.

previousproblem

immediately

tions.Furtherexamplesofthis usefulscheme

e theresistanceRintotwoparallelparts

R.Ifwe callthepartsRiand R2,thenwe

ftheseresistancesisarbitrary.Nowconsider

ssociatedwithRi,andreconvertthiscom-

oltagesourceinserieswithRi, usingagain

ntyielding

ntyielding

nFig.12 butinther eversedirectionascon-

hitwaspreviouslyused.Thevoltagesource

emberingthati,(t)=e,(t)/R,inwhiche,(t)is

eofFig.1, weendupwiththe situationpic-

ecurrenti(t)through theinductance,fora

nthesituationof Fig.1.Weseebythese

euvers(effortlessbecausenodifficultcalcula-

olutionare involved)thatthesolutiontowhat

ea morecomplexnetworkproblemisrepre-


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RLCIRCUIT

rivialvariationin theknownsolutiontoa

idertheproblemofFig.1but thistimesplit

componentsRiandR2 leavingRi+R2=R.

gesourcee,(t)inserieswithRi intoanequiva-

,(t)/Riin parallelwithRi,wegetthe net-

whichi(f)isstill thesame[foragiven e,(t)]as

nsideringthesituationofFig.1.

aderthatsuchtransformationscanbecar-

napplyto RiinFig.17 thesameprocessthat

,etc.),andthattheprocedurecanbe varied

onR.Furtherexploitationisleftas exercises

LCircuit

o thecircuitofFig.18.Herewe havethe

apacitanceCandaconductanceGfedbya

evoltagee(t)appearingacrossthis parallel

sthedesired

iscaseis an

disexpressed

thatof

nwithEq. 1

mofthecircuitofFig. 1,weseeatoncethat the

pecifically,Eq.44becomesinterchanged

gesthefollowingquantitiesinpairs:

eis furtherevidenceoftheprincipleof

d.WerefertotheconfigurationsofFigs.1

ofeachother,andthe reasontheyareduals

ationsfollowone fromtheotheruponinter-


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UNCTIONRESPONSE

concepts

allelconnection

eformasEq.1,we canwritedownthesolu-

fwe leti,(t)bea unitimpulse(itthenhas

=0say,fillingthe capacitanceCwith

voltageto theinitialvalue1/Cvolts),then

from Eq.14asbeing

nlookslikethecurveof Fig.3withtheinitial

tsequalto C/GinsteadofL/R.Thedamping

circuitis G/C;itstimeconstantisC/G

ep,thesolutionhas theformofEq.18.

curveinFig.6with l/Gastheasymptotic

scircuitintoa varietyofotherforms,some

rkout forhimself.

getothercircuit

latingthecapaci-

itthiscapaci-

sCi andC2,

siderCinearest

ecurrentsource^and convertthis

). ..... ..,.

ntoanequivalentvoltage

uivalenceshown

circuitofFig.19 inwhich

hesameas inthecircuitofFig. 18forany


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mpulse,thene,(i)is astepfunctionofthe

ese(t) foraunitimpulsei,(t), wecansay

eofthecircuitof Fig.19reads

2)(50)

ontothenextproblemin orderofdifficulty.

ngallthreekindsofelements:R,L, andC.

uit withFig.21.Equivalentcircuitarrange-

othatinFig.20, whichisfound

ansformation

seriesarrangement,fedbya voltagesource,

briumequationreads

51)

etus chooseaunitstep.Thissituation

tivelythroughconsideringthesourcetrans-

.14wherea voltagesourceinserieswith

intoan equivalentcurrentsourceinparallel

r presentproblemwehaveachoiceof con-

einseries withanyoneofthe threeelements

gehasbeenassumedtobe astep'function,

involvingthecapacitanceturnsoutto be

rwillsee inamoment(hecan thentryoutthe

elfifhe doesn'tagreethatourchoiceis a

tethecapacitanceCwiththe sourcee,(t)

rmationshowninFig.14, wegetthesituation

chthecurrentsourceis Ctimesthederivative

mpulseofvalue C.Nowitshouldberemem-


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UNCTIONRESPONSE

nCh.4 thatacurrentimpulseappliedto a

cesafinitechargeinthatcapacitance.In

associatedelementsRandL,buttheirpresence

nfluenceuponthisinstantaneoushappening

crossthecapacitanceisonlyfinite,andhence

pathwill certainlyhavetobefinite.This

rthecapacitancevoltageinzerotime,andso

nstant whenthecurrentimpulseimpinges

edabove,notin-

enceofRandL.

(whichinstant

thusituationis

celesscircuitof

citanceCfindsit-

ulombsatt= 0(that

. , . , , „t. — x ,

n,is,raisedtoapotentialof 1volt).Thusthe

step voltageto

isreducedtotheproblemof thesimpleca-

eriesRLCcircuit,thechargebeingsuchas

itiallyto1 volt.

orthissimplifiedversionof ourproblem

)

nentialsolution,andtrytentatively

ds

sclearthat

valueswhichforthemomentweshallcall

luedeterminesanindependentsolutionofthe


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ecognizethattheirsumisa moregeneral

eregardedas integrationconstants.These

abletomakethe solution57fittheconditions

nceappearschargedto1 volt;thatis,

mbs.Thecurrentatthismomentisstill zero

entthroughtheinductancewouldrequirean

tpresent.Thechargein thecapacitanceis

nbyEq.57.Denotingthechargeby q(t),

Candi=0.WritingEqs. 57and58for

AiandA2are determined.Onereadilyfinds

esintoEq.57 yieldstheformalsolutionto

ep valuesarefoundfromEq.56.

hesolutionthusfounddependsvery defi-

e pvalues,orrootsof thealgebraicEq.56,

lledthe"characteristicequation."thepvalues

toas"characteristicvalues."Primarilythe

terminedbythevalueofther esistancepa-

56.To illustratethispoint,letus consider

and,tobegin withthatvalueleadingtothe

antcase,chooseR=0.

as tomakethetopplateof thecapacitanceinFig.21

cearrowforthecurrentis suchastomakethe bottom


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UNCTIONRESPONSE

C(61)

ave

p2=—./wo(62)

pairof conjugatepureimaginaries.From

n «0<(64)

0,Lw0= l/Cu0.

asimplesteadysinusoidwith angular

seconddeterminedbytheLCproduct,since

4or 65intheform

plitudeof thesinusoidalresponseisdetermined

etwo circuitconstantsLandCdetermine

boutthesinusoidalresponse,namely,its

referredto asthenaturalfrequencyofthe

dbytheLCcircuit.AsshownbyEq.62 this

minedfromtherootsofthe characteristic

sundamped(thatis,it persistsindefinitely)

sistanceparameterR,whichis thedissipative

Inanypracticalcircuittherewill always

stance.HencethepresentexampleforR=0

unattainablelimitingsituation.Nevertheless,

causeitemphasizesthemostsignificantchar-

cuit,whichisits oscillatorytendency.We

ndreturntothecharacteristicEq.56 inthis


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47

essofnotationitis expedient,inaddition

et

en

cannowoccur,dependinguponthefollowing

,o-«'^ 3

chresultsifc"

ase,definedby

ycase,ifa^ 0butAj^i/^'^r"j1'^

lastisby farthemostimportantpractically,

cussionswillbe substantiallyconfinedtoit.

owriteEq.69in theform

netherootsas

a—jud(74)

ednaturalangularfrequency,indistinctionto

ytobe theundampednaturalfrequency.Itis

otefromEq. 75thattherelationamongthe

oreferred torespectivelyas"overdamped,""critically

mped."


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TIONRESPONSE

sidenticalwiththatrelatingthe threesides

Ifw0is regardedasfixedandais thoughtof

owsthat itvarieswiththe resistanceR)one

arpictureofhowthe dampednaturalfrequency

pedvaluewofor varyingamountsof"damping"

eof a.

ndp2,the constantsAi

60become

howthatpip2 =a2+u*2

eformalsolution57 gives

nwrf<

sameasforthepre-

xceptthatthesinusoidaloscillationisnow

causeofthefactore-at,andthefrequencyof

creasedasaresultofthe dampingeffect

.23.Thequantitya,givenby Eq.67,is

itsreciprocalisagaina timeconstant.

eamplitudeofthesinusoidaloscillation

=0.368of itsinitialvalue;in2/a secondsit

its initialvalue,etc.Thissituationmay

edifonevisualizesthecurvefor e-at—essen-

as forminganenvelopefortheoscillatory

f theexpressioninEq.78.

y Eq.67intothinkingthat thedamping

nRand L,andnotuponC,for theactual

e otherwise,dependinguponwhatelse(such

tacitlythinkingofasbeingheld constantwhile

ew02=l/LC,

ueofw0,ais seentovaryasthe productof


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5onewouldget

foraconstantvalueofwd,a dependsuponthe

scillatorysolutionasportrayedbyany

ns77,78, or79,iti ssometimesofinterestto

aythattakesplaceduringone cycleorperiod,

of oscillationatthebeginningofanyperiod

riod isgivenbythefactor

sfactor,whichis

(84)

ecrement"orsometimesjust"thedecre-

sefultoobservethat1/Aequalsthe num-

whichtheoscillationsdecaybythefactor

dto be"highly"oscillatoryor"slightly

unicationsapplicationsthissituationis of

ty"ofthecircuitbeingregardedas higher

ampingpresent.In theslightlydamped

arithmicdecrementmaybewritten

Vc/L(85)

heQof thecircuitisameasureof itsquality

termsof Qthelogarithmicdecrementreads

cesinwhichanonoscillatoryorsubstantially

sdesiredbutwithapremiumuponthe rapidity

tainotherfixedconditionssuchas agiven


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UNCTIONRESPONSE

e.Acaseinpointistheballisticgalvanometer

onpendulumandhencea mechanicalanalogue

twe arestudyinghere.Althoughthe

ferredtoabove)ismoreor lessthesolutionto

oblem,itisfoundthat anadjustmentwhich

stoanoscillatoryconditionbutverynearto

eisa moresatisfactoryanswer.

e worthpointingoutthatthesolutions

odicandcriticallydampedcasesarereadily

tatedbyEqs. 77,78,or79.Thus,when

udasgivenbyEq. 75becomesimaginary,

te

ther ofthetwoprecedingequationscould

ertinenttotheaperiodiccase,namely,

sinh fit(90)

snonoscillatory,thistimefunctionhasthe

onalpulse.

sefibecomeszero.Notingthat,for small

tforthis caseisimmediatelyobtainedfrom

t thatavarietyofmodificationsofthe

.21maybecarriedout withoutaffectingthe

ough

ource

Figs.12,

paci-

to

orma-

(c)

rtthis

ntyieldingcombinationintoavoltagesource

0and21.e,(t)= l/Cjtimestheintegralof

cuitof

epfunctionofthe valueC/Ci,thecurrent

t determined.


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RLCCIRCUIT

ociatethevoltagesourcewiththeinductance

cearrangementaccordingtotheequivalence

totheadditiveparts1/Li+ 1/L2,associate

nvert

rangement

Figs.20and21.

t has

ig.25

again

have

altoa

i/L

tor

me

ir-

ughuse

ommonwiththeprincipleofdualitywhich

thefollowingarticle.

LCCircuit

alofthesituationgivenin Fig.20isthe

ninFig.26.InsteadofR,L, andCinseries

uals(G,C,and L)inparallel.Insteadof

agesource

current

ofbeing

urrent

allingthe

theparallel

dence

actually

shown

rticleisrevealedin theequilibriumequation

rentequilibriumascontrastdto thevoltage

Eq.51)

.

ofthedualquantitiesin thepairs—voltage

dconductance,inductanceandcapacitance—

aspreciselytheformof Eq.51expressingthe

Fig. 20.

refully,inregardto duality,thatthis

lone.That istosay,it iseverybitas ap-


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TIONRESPONSE

etworkofFig.20isthe dualofthatin Fig.26.

ardedasthe givenoneorasthe dualofthe

thediscussionsof thischaptertobegin

eries,whencethedualthingsconnectedin

situation.Wemightequallywell have

angement,whencetheseriesarrangement

outasbeing thedualone.Inother words,the

thehabitofthinkingthatdual meansthings

happenedto startourdiscussionswitha

ldn'tverywellhavestartedwithbothtypesof

y).

existingbetweenthetwonetworksinques-

own thesolutiontothepresentproblemat

ddiscussiongivenin theprecedingarticle.

hecharacteristicvaluesofthis systemare

sibilityof overdamped(a>w0),critically

latory(a< u0)cases.Particularlyforthe

is case,assumingi,(t)tobea unitstep,

77,78,or79, andreads

udt(97)

nisidenticalwiththatalreadygivenand need

maybe recognizedasapplyingtonumerous

mentsderivablefromtheoneshowninFig.26

ormationsasillustratedbefore.Forexample,

yieldsthe samee(t)asthatin Fig.26for

isanimpulseofvalue L,thesolution

ctly;if e,(f)isaunit impulse,thene(t)is


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TRARYINITIALCONDITIONS

97, multipliedby1/L;thatis

26isfirst splitintotheparallelcomponents

ourceconversionappliedtoi,(t)incombination

howninFig.28 inwhiche(t)isagain given

ntyieldingFig.28.Alternatecircuitarrangement

6.yieldingthesamee(0as inFig.26.

seof valueLi.Furtherexploitationofthese

xercisesforthestudent.

yInitialConditions

noftheresponseof severalsimplecircuits

ageorcurrentstimulus,it istacitlyassumed

hecapacitanceandnocurrentinthe induct-

renceofthisexcitation.Sincetheinstantof

redexcitationiscommonlyreferredtoasthe

ythatin ourprecedinganalyseswetacitly

ons,"orthatthecircuitto whichthegiven

allyat rest.Inpracticeitsometimesisneces-

seofa circuitthatisnotat restwhenthe

.Wewishtosay afewwordsnowaboutthe

onmaybe dealtwith.

uitthatis notatrest,wemayinfer thatit

dingtosomepreviousdisturbance.Thatisto

conditionofinitialunrest,wearedealingwith

essofcompletingsome"unfinishedbusiness."

iedwhilethecircuitisthus inthemidstof

ows,becauseofthelinearityof thedifferential

etwork'sequilibrium,thatthenetbehavior

addingtogetherthe"unfinishedbusiness"

hefreshexcitation,thelattercomputedas


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TIONRESPONSE

iallyatrest.* Itthusismerelynecessary

outdescribingmorespecificallythe"un-

off-handimpressionthatitis necessaryto

storyofthe circuitinordertobe inaposition

tersomespecified"initialinstant";and,since

yinsomecaseshavebeenrather complex

one doesnotcontemplatethesecomputations

ameofmind,nor doesitseemprobablethat

onesimpleformulacapableofdealingwith

however,thatwecaninfactfindsuch a

eneralformula,forthefollowingreason:

chthegivencircuitmayhave beensubjected

tialinstant"is completelysummarizedbya

nthecapacitancesandthecurrentsin theinduct-

tfollowsfromthe factthatthebehavior

itialinstant iscompletelyanduniquely

esofthecapacitancechargesandinductance

usthe foregoingdiscussionhasshownthat

L circuit(Fig.2)for(>0 isfixedbythe

=0(Eq.14 istheresultfor i(0)=l/L).

eseriesRC circuitfort>0 isfixedbythe

0(inEq. 47thecapacitancevoltageisgiven

argeis 1coulomb).TheseriesRLCcircuit

>0in termsofknownvaluesofchargeand

showninfurtherdetailpresently).

thenetworkatt =0is adequatelyde-

e initialchargesandcurrents;itis notneces-

uescameabout!Althougha givensetofvalues

result ofmanycompletelydifferentbehavior

nstant,thebehaviorthat thissetdetermines

epatternbecausethesolutionto thepertinent

uilibriuminvolvesasmanyintegrationcon-

dentinitial chargesandcurrents,sothatthe

stantsuniquely,andnothingelsecanexertan

olution.Thispointis nowfurtherillustrated

scussedin Art.5.

nthe discussionoftheRLCcircuitwhere

rentandcharge(Eqs.57 and58)areob-

lEq.52.Thereason werefertotheseas

exampleofthissort.


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TRARYINITIALCONDITIONS255

eyarenot yetexplicitrelationsforthe

relyrepresentthesequantitiesin functional

onconstantsAiandA2appearingin themare

two unknownconstantsareinvolved,two

torendertheformalsolutionsexplicit.

aychoosetwoarbitrarilyspecifiedvaluesof

antsoftime,or twovaluesofthechargeat

smorecommonlydone,wemayspecifyvalues

argeatt=0. Wecallthesevaluesthe"initial

erminethestateofthenetworkat t— 0.

Eqs.57and58for t=0,we haveforthe

rationconstants(inplaceofEqs.59)

(0)(99)

2Pi?(0)

aluespiandp2are conjugatecomplex,we

gatecomplex.ByEqs.74and75wefind

)=juoe'*

ja)=—ju0e~3<t'

(wd/«0)(102)

ationofEqs.100, wehave

d

d

](104)


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UNCTIONRESPONSE

give

0)cos(udt+ct>)- u0q(0)sino^e-0'

os(udl- ct>)+— sinudte-°'

madeuse ofthefactthatthe termsinEqs.

omplex;whencetheirsumis expressibleas

erone.Thestudentshouldverifytheexpres-

substitutionoftheresults104and 105in

hthe manipulationofcomplexquantities,

onentialfunctionwithcomplexexponent.

dexplicitlythe currentandchargeina

ntermsof theirvaluesatt= 0.Thus,

otheidentitiesi(0) =i(0)andq(0) =q(0),

,it beingclearfromEq.102thatcos 4>

reusefulin thattheyaretherepresentation

sthattheRLCcircuitmay finditselfinthe

t=0 afreshexcitationisappliedto it.

fthisunfinishedbusiness,onewrites Eqs.106

equaltotheappropriatevalues(thesemust

andthenaddstheresponsedue tothefresh

medrestconditions)toobtainthecomplete

ethatEqs.106 and107givetheresponse

a numberofspecialexcitationfunctions.

eresponseof thiscircuittoanappliedunit

veaccordingtothediscussiongivenearlier

yestablishesacurrentin theinductanceof

eedmerelyconsiderEqs.106and 107for

havetheresponseappropriatetothis

own(throughconsiderationofFigs.20and

epvoltageisequivalenttostartingfroman

eq(0)=— Ccoulombs.HenceEqs.106and

— Cyieldresultsappropriatetothis case,

parisonwithEq. 77.

ustorecognizethattheprocessof taking

ntoaccountinatransient-netwoikproblem

teway.Thustheexistenceofacurrentin an


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ETRANSIENTRESPONSE257

valenttoinserting avoltageimpulse(of

urringatt= 0)inserieswith thisinductance,

argein acapacitanceatt=0 isequivalent

se(of appropriatevalueandoccurringat

nce.Inotherwords,anysetofarbitrary

smaybereplacedbyan appropriatesetof

sesourcesconnectedintothenetwork.Super-

yproducedresponsesandthatdue tosome

mputedforinitialrestconditions,yieldsthe

discussionofnetworkresponsethattacitly

onditionsisneverthelesssufficienttodeal

bitraryinitialconditions.

ETRANSIENTRESPONSE

EE-ELEMENTCOMBINATIONS

ource


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ETRANSIENTRESPONSE259


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UNCTIONRESPONSE

l/VZc

inuot

O'


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HE

a^C,L

cosw0<

~"sinwdt


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TIONRESPONSE

os1—

su,o<

cessaryonlytoinsert easamultiplier tothere-

sers

Fou_i(0inseriesi,{t) =CF0Uo(<)iQparallel

parallele,(<)=L/omo(<)in

t inthecircuitsolve bysuperposition.

low,determinee(t)andi(<)(a)for t',(<)=uo(<),(b)for


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ove,determinee(<)andi(<) (a)fore,(<)=u0(t),(b) for

rminee(0andi(t).

nee(0andt(0.

erminee(0and t"(<).

erminee(<)andi(0.

ow,Rirepresentsthe leakageresistanceofthecon-

Whatistheequationforthe chargeonthecondenserasa

witchK isclosed?

itsinProbs.1to 6inclusivearepotentialduals?Using

uranswerstotheseproblems.

<t <L/Randisopenduring theintervalL/R<<<«>.

forthevoltagee(0validfor 0<t <L/Randagainfor


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UNCTIONRESPONSE

lychargedtoEvolts,is suddenlyshuntedbyi

heknownsolutionforthe currenti(<),computethe1

mthe integral

alstoredenergyin thecapacitance.

=1henry.R =100ohms.Thestoredenergyic 1

erminet(0fort> 0.Computethetotal

sofa capacitanceCi,aresistanceR,asecond

hthe switchopen,Ciischargedto1 volt.Theswitci

ethecurrentinthe circuitandthevoltagees(t)acros..

+ I/Cj.Determinetheinitialandthe finalenergv

2,andtheenergylost inRfromanintegrationof vR

limitingprocessR—>0,andstate theenergyrela-

mit.

scopefortherecordingoftransientsit isnecessaryths:

sbe suppliedwithavoltagethatis alinearfunctionc:

nginterval.A simplewayofgeneratingareasonabh.

stypeofvoltagefunctionis essentiallythecircuitdescribed

thevoltagein question.

nearityrequired,supposeitis stipulatedthatej(/)shali

t =0byat most5percentthroughoutaninterval

~S farad,andCiisat least100timeslarger.Sensi-

oscilloscopeindicatethatej(<)shouldvaryfromzero to

propriatevalueofthe resistanceRintermsofthe inter-

tialvoltageofthecapacitanceCi.

impresseduponaseriesRLcircuitinitiallyat rest

dintheinduc-

orthe in-

<t<2L/R,

impressed

ally atrest.

dinthecapaci-

orthe in-

t<2RC,

theaccom-

Siscausedto snapbackandforthbetweenpositions1

equaltothetimeconstant,thusalternatelyimpressingthe

vingthecircuitcloseduponitself, (a)Determineandplot

vior,(b)Startingfromrestconditions,supposetheswitch

fortheinterval L/R+iseconds,andthensnapped


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dabove.Whatshould bethetimeincrement5 inorder

ensuesafterthe

?

energysup-

eriod,andcheck1 iJi

.© [i,(t) iW

ppliedtoase- "3

antcurrent

mplitudeof10'{

of10* cyclesperProb.20.

uesofLand C.

f voltageoccurringacrossLandC separately?Sketch

rent(neatly)for severalcycles.

rminee(<)andt'(<).

a,e2,• ••arepotentialsof theindicatednodeswithre-

m.Utilizingthe resultsofProb.20,determineallindi-

tsfort, =uo(<)-

sourcei,impressedbetweendatumandanyofthe

veralwidelyseparatednodes,andnoteparticularlythe

alcomponentsofvoltageandcurrent ineachcase.Can

kproperty?

re,thecurrentsourceis impressedacrossafractiona

ductanceL. Ifi,=u_i(<),determine«i(<),«c(<)>>(<) for

ilfluxlinks allturns.

rieswithelementsLand Chastheformof arectangular

ermineandplotthe resultantcurrent(a)fori equalto


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UNCTIONRESPONSE

ultant oscillation,(b)forJequal toawholeperiod.Lo1

gofthe pulse,andextendyourplotsoverthe interval

4,supposethevoltagesourcehas theformofthepulse

ingequalpulseamplitudes,statesomecombinationsof

atwillyield amaximumoscillationafterthepulsetrain.

hatwillyielda restconditionafterthepulsetrain.

ketch,£is aconstantresistancelessvoltagesourer,

nappedinstantlyfromone positiontoanother,andDis

offerszeroresistancetocurrentin theleft-to-rightdirec-

to currentintheoppositedirection.AssumeE =10

=10H(second)2.Theswitch isfirstmovedtoposi-

0microseconds,

position2.

thecapacitance

odofsolution

ilartothatgivenin

lyofenergyre-

oec/E(where<<

ctionofL,C,

ttheswitchis

nappingitinto

ibilityof thismethodoftransformingadirectvoltage

.

rcuit ofProb.26bereplacedby aresistanceRof

ditionsas thosestatedinthefirst paragraphofProb.

sulting currenti(<)

sthe capacitance.

ra timeinterval

oscillatoryre-

the"dec-

r howmany

oscillationdecayed

Whatshouldbe

scillationdecays

0 cycles?

27,compute

inthe resistance

gyinitiallystored intheinductanceL.


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sed.FindthevalueofCnecessaryfor criticaldamping

Computethetransientcurrentin theLCbranch.

ircuit,theswitchKis suddenlyopenedatt=0 when

amperesanddi/dtequals1amperepersecond.Findthe

<)immediatelybeforeandagainimmediatelyafterthe

ssumefirst thatt',2ande,are bothzero,butthati,i

netheultimatesteadyvaluesofcurrentand voltagein

tevaluesfortheexcitationfunctionst,2ande,suchthat

boveareimmediatelyestablished;thatisto say,determine

ausethetransientresponseinitiatedby i,itobeabsent.

trest,andtheswitchclosesat t=0.Whatmust i,

eensuesimmediately,andwhatisthissteadystate?

ircuit,bothswitchesareinitiallyopenandthe capaci-

Sicloses,and t'(<)beginstoincrease.Ataninstant

lueto <E/R,Sicloses.Finde(t) for<>0 intermsofit

d alsothecurrentii(t)throughthe inductancefromthe


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TIONRESPONSE

me afterStcloses,showingallsignificantvaluesandchar-

ons.

tedualofthesituationgivenin Prob.32includingthe

m,problemstatement,anditssolution.

tedualofthesituationgivenin Prob.33,includingthe

m,problemstatement,anditssolution.

networkthesourceisa voltageimpulsee,(i)=10uo(0,

he currentthroughthe4-henryinductanceis

8c os(9i+10°)amperes

sshownbelow.

namperes)ofthe dualisnumericallyequaltothe voltage

originalnetwork,(a)Givethenumericalvalues(in

ofallthe circuitelementsshowninthedualnetwork,

crossC.(c) FindthecurrentthroughtheresistanceRi-


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he voltagesourceisaunitstep, e,(t)

— 0are:no

e,butacurrent/o

ductanceinthe

cuitofFig. 20

itationdirectlyfrom

gthat theper-

hencethecom-

hecomplementary

orce-freeequilib-

vingforAiand

gral58,considerEq.57 anditsderivativefort =0,

t)for t=0 fromtheoriginalequilibriumEq.51by

ftert— 0,e,— 1whilethecurrentandcondensercharge

ehelpofEqs.74 and75,checkthevaluesgivenforAi

usproblem,observethatthesuccessivederivativesof

sideredatthe instantimmediatelyafter<=0will yield

f higherinitialderivativesforthecurrentso thatone

eroftermsin aMaclaurinexpansionofthisfunction.

ermsintheMaclaurinexpansionoft'(0,and checkagainst

nofEq.78.Contrastthismethodof obtainingthefunc-

gconventionalsolutionofthedifferentialequationplus

onstants.

Fig.20 foraunitstep voltageexcitationthrough

rgeq(t)ratherthanthe meshcurrenti(t)tobe theunknown.

mthe equilibriumequation

egralis notzerobutequalsa constant,asdoestheexcita-

e.Thetotal solutionconsistingofanonzeroparticular

taryfunctioninvolvestwoconstantsofintegrationas

edfromstatedinitialvaluesofchargeandcurrent follow-

minginitialrest conditions,obtainthecompletesolu-

ifferentiateitto findthemeshcurrent,andcheckyour

unitstep.Forinitial restconditions,finde(0through

oftheresultgivenby Eq.97forthe stepresponseofthe


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s

ate

a PredominantPartintheStudyof

nofelectricalphenomena,theearliest

(apartfromvariouselectrostaticgenerators

trole)werebatteriesofonesort oranother.

esourcesofsubstantiallyconstantvalue.

ewasintroducedasamorecopiousand reliable

toproducean essentiallyconstantvoltageat

estexperiencewithelectricalnetworkre-

texcitationfunctions;and,sincetheresulting

tant,suchcircuitsarereferredtoasdirect

ieflyasd-ccircuits.Thediscussionof their

erone ontheagendaofmostelectrical

hereasonthatthemathematicalprocessof

nse(solongaswerestrictour attentiontothe

stwhentheexcitingforceis aconstant.

ssed,wefindthat theconstant(whichis

ematicaltimefunction)wasgraduallydis-

imefunction,namely,thesinusoid.We

infinitepossiblearrayoftimefunctions,the

the onethatshallforeverbeking andruler

ktheory.Theanswercannotbegiven ina

ereasonsforthis momentouschoicebefully

roceededalongwayinto thestudyofnetwork

bothsatisfyingandanalyticallyhelpfulto

ostcogentreasonsfortheimportanceof

sintheprocessof becomingacquaintedwith

kconsistingoflinearelementsisexcited

rcethat isasinusoidaltimefunction,the


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CHAPREDOMINANTPART271

esandcurrentsinallparts ofthenetworkare

nctions,differingfromeachotherandfromthe

tintheirrespectiveamplitudesandtime

haveastructureofunlimitedcomplexity;

mplexlumpedstructures,geographically

ces,andinterconnectedbytransmissionlines

;still thevoltagesandcurrentseverywhere

mefunctionswhichinessenceareidentical

l.Nootherperiodictimefunctioncanclaim

stratethetruthofthis statementonceone

mof anelectricalnetworkisdescribedbya

nwithconstantcoefficientsorbyasetof simul-

ort,forit isreadilyappreciatedthatthe

anequationis asinusoidwhentheright-hand

xcitationfunction)issinusoidal.Thesedetails

Meanwhileitismoreusefulto pointoutother

sinusoidalfunction.

eculiar—yes,almostuncanny—aboutthe

givesitthepropertyof remainingunaltered

vingthesameperiodbut anyamplitudeand

t. Thatistosay,if oneaddstogether(or

idalfunctionshavingarbitrarilydifferent

es,theresultingfunctionhasthe samesinu-

tuentprovidedonlythattheir periodsare

t thesumofanynumberof sinusoidsof

plitudesandphasesyieldsagain asinusoid

plitudeandphase ofthisresultantsinusoid

onthevariousamplitudesandphasesof the

eringintothesummation,buttheshapeofthe

usoidal.Noother periodicfunctioncan

ther.

asfurtheruniquecharacteristics.The

pe—ofasinusoidisagain sinusoidalinform;

egral.In factonemaydifferentiateorin-

yoften;theresult stillhasthesame shape

with.Anyonewhohas someexperience

ntationoffunctionsandtheirderivativesand

eraltheprocessof differentiationtendsto

tiesintheformofa givenfunctionwhileinte-

fectofsmoothingoutthe function.The


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HESINUSOIDALSTEADYSTATE

tsshapeis invarianttointegrationordiffer-

fteneitheroperationmayberepeated.

inthelasttwo paragraphsareindeedthe

ctthatthe responseofanetworktoa sinu-

wisebesinusoidal,forthesumof termsin

onwithconstantcoefficientsyieldsasinusoid

usoidof thesameperiod.

onfortheimportanceofthe sinusoidal

yisdueto theexistenceofatheorem,first

maticianFourier,totheeffectthatanyperiodic

nlimitationsthatarenotimportantat the

matedwithafinitebutarbitrarilysmall toler-

perpositionofa finitenumberofsinusoidal

oximatingfunctionisknownasthepartial

oughthe simpleartificeofregardingthe

nto bearbitrarilylarge,oneisable toapply

ofrepresentationtofunctionsthatdo not

usobtaintheimportantresultthat anytime

haveto dealwithinour practicalnetwork

orsteadystate)canberepresentedasa sum

sstatementmeansthatthe sinusoidisthe

anythingelsecanbeconstructed,andthat

f anetworktoanyformofexcitationfunction

alresponse.

kresponsetononsinusoidalexcitationfunc-

wemayormay notavailourselvesofthe

ntresult,for itissometimescomputationally

byothermeans.Nevertheless,theexistence

estimabletheoreticalimportancesince,in

ardthe investigationoflinear-network

teadystateas nolessgeneralthanwouldbe

ehaviorwithperfectlyarbitraryexcitation

gnificanceofsinusoidalfunctions,itis

nusoidis nature'sbuildingblockalso.Thus

simplestphysicalsystemcapableofsus-

usoid.Thesimple pendulum(exceptfor

utessimpleharmonicorsinusoidalmotion.

itcapableofoscillation(thecombinationof

citance)executessinusoidalbehavior.The

earnetwork,howevercomplicated,is

onofsinusoidsor exponentiallydamped


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ONOFSINUSOIDS273

evealedtoplayadominantrole inournet-

atweacquaintourselvesinordinatelywell

sticsandthe manifoldwaysinwhichitmay

ented.Thefollowingarticleelaboratesupon

nofSinusoids

of electricalnetworks,complexformsare

ntthantheequivalenttrigonometriconesfor

dy-statefunctionssuchasvoltagesandcur-

urallyleadtotheexpressionofimpedances

exformand,therefore,toa morecompact

fexpressingthepropertiesofthe circuits

tion* isexpressedbythe identity

x= — (e*x -e~ ix )(2)

as

>)(3)

onehas

'-W]

exvoltageamplitudeas

sa symbolforThereaderisexpectedto be

ofcomplexnumbersandwith functionsofacomplex


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everyinstant,theconjugateofthefirst.

usedtoindicatethatonlythe realpartofthe

tyistobetaken,onemaywrite

oltagefunctioniscontainedintheexpression

xcharacteroftheamplitudeEasshownin

husseentobe alternativeexpressionsfor

dbeobservedin connectionwiththeform

uentsEe?atandEe->atseparatelyhaveno

onlytheirsumthat iscapableofrepresenting

helessonemayinquireastowhat thenet-

a "voltage"functiongivenbysaythecon-

seduponacircuit.Toil lustratewitha

rcuitbearesistanceRin serieswithanin-

quilibriumreads

is theresponsesought.

ta functionoftheform

substitutionyields

etting

y

mpedanceofthecircuit.

shiononefindsthat,if thecomplexcon-

dintoEq. 9inplaceof Ee,'at,thenEqs.10

5)


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ONOFSINUSOIDS275

.

superpositionwhichstatesthat,since the

separatesolutionsaresuperposable,onemay

andside ofEq.9is replacedbythevoltage

ngcurrent (aphysicalcurrentthis time)is

of expressions10and14,thatis,

ivelygivenbyEqs. 12and16.

velybewrittenas

ultingcurrentforthephysicalvoltagefunction

Eqs.3,7,or 8maybeobtainedthrough

exconstituentEg!"'alone,andsubsequently

ftheresultingcomplexcurrent(Eq.10).

rstood.thatonlytherealparts oftheindi-

saretobeconsidered,it ispermissibleto

ationsbywritingforthe voltageandcurrent

ondingcomplexconstituents,thus,

ntand voltageamplitudesarerelatedthrough

hipis physicallydeterminedbythecircuit,

are,incidentally,observedtobecompletely

impedanceZasgiven inthissimpleexample

ntforms3 and8,itis seenthatEq.18

itteninthe trigonometricform

20)


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ce,onehas

e phaseangleoftheresultingcurrentis

8byvirtueof thecomplexcharacterofthe

teady-statebehavioronly,itis ordinarily

exponentialfactore^ut.Theonlysignificant

metricalin-Fio.2. Thecomplexcurrentvectorin

ingtheEq. 18atatime t>0.

rrentvector

whicharesimplyrelatedbythe complex

heeffectivenessofthecomplexnotation

cumspectionwithwhichonemaydealwith

ike frequencythatdifferintheirampli-

es.Thiscircumspectionstemsfromthefact

accomplishedthroughavectorialaddition

es.

one,it isessentialfirstto grasptheso-called

asinglesinusoid.Herethesketchesin Figs.

atingtheinterpretationofEq.18in thecom-

thecomplexfunctionIe?atfor t=0,while

nisdrawnasit appearsataslightlylater


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ONOFSINUSOIDS277

ans.ThusIis thiscomplexfunctionat

sequentinstantthefunctionisthis same

roughanangleof utradiansinthe counter-

on.Toindicatethatthevector/ rotates

elocityuradianspersecond,it iscustomaryto

ctor asmallarrowlabeledu,pointing inthe

dalfunctioni(t)isat anymomentequalto

g/vector;thatis, i(t)equalstheprojection

?"'upon therealaxisofthe associatedcom-

pretationofFig.1 pertinent(a)tothesine functionand

wiseindicatedinthetwofigures.It is

ct>,theangleofthe complexamplitude/

of*(<) att= 0.

howthe complexrepresentationofFig.1

ci> thatareofspecial interestbecausethey

oi(<)beingasine oracosinefunction.The

0,and becomespositiveimmediatelythere-

dingpositionofthevector /mustbesuch

pontherealaxis,but willhaveapositive

tate.ThepositionshowninFig. 3(a)fulfills

nefunctionhasits maximumpositivevalue

rrespondingvectorJisseen tocoincidewith

ownin Fig.3(b).

ehelpfulin theinterpretationofthegeneral

gest thatthelattermaybe regardedasa

andcosinecomponents,andthus theyyield

orthefamiliartrigonometricidentityapplied

cosct> cosut— | / | sinct> sinut(26)


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calin-

.4.

seen

sum

ofthe

nce

mponent,

positeto

sinusoidthat rfI r3(r)andheQce

nto sine.

q.28).correspondstoa negativesine

sAand

realandimaginarypartsof I,asis ex-

/ A(31)

hemagnitudeandangleof thevectorIfrom

form28.

oindicatehowthe additionofsinusoids

rriedout. Supposewehave

(«<+

s(w<+<h)

um

""]=|11 cos(at+0)

sequalstherealpart ofthesumoftwo com-

hat

]

rialadditionis,ofcourse,implied.


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ONOFSINUSOIDS279

fthesinusoidsin Eqs.32and33is written,

Eq.28, as

ve

ypartsoftheresultantcomplexamplitude.

torialadditionimpliedin Eq.36.One

ding togethercorrespondingcomponents

nd I2soas

theresultant

onemay

hroughcon-

of aparallel-

andI2,asin the

mechanics.

ved,how-

nprinciple,

oregoing

onnection

etheuseof

utlinedis

onssuchasad-

entiation,and

eoperationsencounteredinthesolutionof

nswithconstantcoefficients.Tostatethe

maysaythat, sincethefourabove-named

eduponcomplexexpressions,arecommutative

onoftaking therealpart(as,for example,

lstherealpart ofthesum,orthe derivative

alpartof thederivativeofacomplexex-

iththecomplexexpressionswhilesolving

andtaketherealparts afterward.

nergyrelationshipswhichinvolvequadratic

edureisnotpossiblesinceit isnottruethat

ofacomplexexpressionisequal tothesquare

-

sed

ough


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y,it isnotcorrectto assumeinconnection

t

nof squaringisnotcommutablewithtaking

aroperationsmentionedabove.

tosquarethe entireright-handsideasin

alentform7,towrite

(42)

ut+2EE) (43)

<](44)

ewritten(throughuseofEq. 5)

s(2ut+2*)(45)

eproductof avoltageandacurrent in

thesimplifiednotationin19, itisnot correct

ntof

f theexpression46betaken.

tationinEqs.19 isusefulindealingwith

ory,thestudent shouldnotlosesightof the

suse,or therestrictionsthatarethus implied.

omplexexpressionswillbecomemorein-

theuse ofcomplexnotation,afewadditional

ghttothe student'sattentionatthistime.

ure,giventhecomplexnumber


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ONOFSINUSOIDS281

orm

dfor,a tendencyisfirsttoput thisgiven

dthen apply49:thatis,throughration-

btainfrom50

bd+ j(bc-ad)

Eq. 49

)2

edureisneedlesslylong.It ismoreexpedient

deofthequotientoftwocomplexnumbersis

eirmagnitudes.Thusthemagnitudeofthe

wnat onceas

2 and53areentirelyequivalent.Themethod

ncursawasteof timeandeffort,butfre-

racteristicsoftheresultantfunctionand

eedlesslydifficult.

eformationofthe magnitudeofaproduct

-bd)+ j(ad+be)(54)

2)(55)

+be)2(56)

usuallypreferableandmoreeasilyformed.

beemployedin morecomplicatedsitua-


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ownatonce asthesquarerootof

a complexexpressionitislikewiseun-

estandardform 48.Thustheconjugate

ample,is simply

pedanceConcept

of Fig.6whichconsistsofa sinusoidal

swith theelementsRandL.The problem

the

sourcehasbeen

ora sufficiently

he transientini-

insertionhas

ue.It is

rsteady

texcitedstatethatis ofinterest.

llustratethe relativeeffective-

ndealing

the presentproblemfirstwithouttheuseof

hit. Thussupposewewrite

tandu =2wfisthe radianfrequencyofthis

uationofthecircuit reads

ststhattheultimatebehaviorofthe current

)

tosupposethattheresponse,likethe source

idal,althoughits amplitude/andtime

excitationmayhaveothervalues.FromEq.62

)


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EIMPEDANCECONCEPT283

61 gives

s(ut+ ct>)]I=Ecosut (64)

solvedforthevaluesof /andct>.Aprocedure

lythetrigonometricidentities

s</>+cosutsinc t>

ct>— sinw<sinct>

ten

sut — /(72sin#+ Lucos<£)sinut =Ecosw<

— 7?]cosut— /(72sin+Lu cos#)sinut =0

8)

ll valuesofthetimet onlyifbothcoefficients

havefromEq.67

E(69)

70)

yields

ofthe currentisdetermined.

well-knowntrigonometricrelations,


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mplitudeofthe current

onstitutethe solutiontoourproblem.

eproblemusingcomplexnotation,writing

plied,andE mayberealsoas tohavethe

to60;orE maybecomplexifwewishfor some

ase ofthesourcearbitrary.Sincephase

citlyincomplexnotationit isjustas easy

eads

algroundsthatthesteady-statecurrentwill

te,/

xcharacterofInowcontainsthetime phase

citmentionthereof.Differentiating79we

assumedsolution79into78yields after

ctore?atappearingonboth sidesofthe

efor/, getting

s

obereal (soastomatchthe situationpre-


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IMPEDANCECONCEPT

metricmethods),wehave

nd 72respectively.

pectnatureofthe complexmethodascon-

coneis striking,eveninthisverysimple

smoreelaboratenetworks,thecomplex

pactandalgebraicallystraightforwardchar-

tricmethodrapidlybecomessoinvolvedas

olutionsin termsofitpracticallyimpossible.

ethodpermitsaninterpretationof oursolu-

tialsimilarityto thesolutionofd-ccircuits

.Thus,ifwedefinethequantity

ries RLcircuit,Eq.82reads

cterof Ohm'slawrelatingvoltageandcur-

nt.Theseries RLcircuitobeysessentially

veto itsvolt-ampererelationinthesinusoidal

detailedrefinementthatthe quantitiesE,I,

m'slawrelationarecomplexquantities

king accountoftheimportantcircumstance

and voltagemayingeneralbedisplacedin

5 initspolar formas

byEqs.83 and84maythenbewritten

0)

dthesource voltageashavinganarbitrary


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gitsamplitudeE as

onlyin themannerindicatedby

eisthe voltagemagnitudedividedbythe

hilethecurrentphaseisthevoltagephase

e.Thissimpleruleholds forallnetworks,

eonlythingthatchangesasweproceed to

circuitsis thedetailedformofthe impedance

uit parametersanduponthefrequency.In

exploitthecircumspectnatureofthe complex

vegraphicalinterpretation,asisdiscussed

nceinthe ComplexFrequencyPlane

tatthispoint inourdiscussionsandreflect

nceofthe steady-statesolutionthatweare

to thetransientbehaviordealtwithinthe

recalledintheconsiderationoftransient

dyingthemannerin Whichacircuitbehaves

alstimulussuchasheavinga packagedcharge

uit islefttoits owndevices;thatistosay,

ecuteitsownnaturalbehavior—abehavior

own.It wasobservedthatthisnatural

ntforms,dependinguponthegeometryofthe

ements(R,L, orC)thatare containedinit.

henaturalbehaviorwasseento beasimple

RLCcircuitit takestheformofa decaying

enerateintoanonoscillatorycharacterfor

alues.

haviorisdescribedanalyticallybyoneor

formAept, inwhichthenatureof thepvalues

re-eminentin settingthepatternofthis

realvalue,suchas —R/Linthecaseof the

edecay;apair ofconjugateimaginaiyvalues,

fCh.5in thediscussionoftheLC circuit,

ation;apair ofconjugatecomplexvalues,as

h.5forthe RLCcircuit,leadtoa damped

,incidentally,thatrealpvaluesandreal


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DANCEINCOMPLEXFREQUENCYPLANE287

lwaysarenegative,thusyieldingexponential

wardzerowithincreasingtime.This cir-

ticofpassivecircuits(suchwithnointernal

withpositiverealpartswould producetime

yincreasewithtime—aconditionthatclearly

sof physicalreasoning.

me beingnotconsiderthetransientor

omplexcircuits(involvingelaborategeometri-

domdistributionsofR,L,andC elements),

mthediscussioninArt.2 ofCh.5that this

scribedbyasumofterms oftheformAept;

worksmerelyrequirealargernumberofsuch

aluesandp values.* Thuswemaysayquite

behaviorofanylinear passivenetworkmay

dition ofdecayingoscillationswithappropri-

ecay,initialmagnitudes,andtimephases.

ofdecayaredeterminedbythe pertinent

are complex),whiletheinitialmagnitudes

minedbythe pertinentAvalues(whichlike-

x,butunlikethe pvaluesdependuponthe

kanditsmodeofexcitation).

allcharacteristicpvaluesascomplex,and

or purelyimaginaryonesasdegenerateforms.

ndadditionallyhelpfultoacceptthenotion

dregardthep valuesassuch.Theimaginary

cyisthus theactualradianfrequencyofthe

on,andtherealpartis itsdampingconstant.

asthoserepresentedbyRLor RCcircuits,the

esure havezeroimaginaryparts,andthe

ons"maynotoscillateatall,butit isnever-

eferto suchcharacteristicvaluesascomplex

generatenessaffectsonlytheirparticular

rethesameasthe onesthathavecomplex

ay inregardtonetworkbehavioristhe

tthepresentdiscussionis leadinguptois a

thesecomplexnaturalfrequenciesorchar-

orknotonlyplaya dominantroleinfixing

ehavior,butalsodeterminethecharacterof

behavior.Infact,asourstudy ofnetwork

mesincreasinglyevidentthatthecomplex

esemattersisgivenin Ch.9.


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earsystemcompletelycharacterizethatsys-

sientandsteady-stateconditionsalike(except

alefactorswhoseindependentadjustmentis

transientbehaviorandsinusoidalsteady-state

uniquelyrelated,onetothe other—acircum-

einproblemsofanalysisand ofsynthesis.

steady-statebehavioritshouldfirstof

completelycharacterizedbyasinglequantity—

itinquestion.Thusin asinusoidalsteady-

venexcitation,saya voltageoffixedampli-

se,andweseekthecurrentthatis forcedto

liedvoltage.Observethatthecharacteror

seis herebeingdictatedbythesource;the

obeythewishesofthisdictator.Thesource

quency,andsotheresponsemust beasinusoid

smuchfollowsfromthelinear characterof

overningtheequilibriumofthecircuit.The

canexertuponthe responseistocontrolits

relativetothe amplitudeandphaseofthe

liscompletelyandcompactlyeffectedthrough

ertoEqs.85and 86ofthepreviousarticle

thecaseofthesimpleRL circuit).

steadystate,thenature oftheresponseis

ytheexcitation,thecircuitis seentoinject

ponsethroughthe effectithasuponthe

e"go-between"asitwerethat placatesthe

d thenetworkontheother,andmutually

dbemorenatural,then,than tofindthatthe

pendspartlyuponsomethingthatis charac-

artly uponsomethingthatischaracteristicof

ethings"are(asthereadermayguessfrom

mplexfrequenciesofthesourceand ofthe

equency"ofthesourceis apureimaginary

ethisdeviceiscalleduponto produceanun-

mplexnaturalfrequenciesofthenetworkare

rementthattheyhavenegativereal parts.

evealedtobeafunctionof allofthesecomplex

plexquantitywhich,asinEq. 86,relatesthe

entamplitudescharacterizingthesteady-state

tersisgreatlyenhancedthroughgraphical

doneintroducesthenotionof acomplex

rvesasameansfordisplayingall thecomplex


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ANCEFUNCTIONSFORSIMPLECIRCUITS289

alsoasapotentmeansfor recognizingby

entialcharacteristicsoftheimpedanceasa

ies.Detailedillustrationsofthesematters

oftheRLcircuittreatedin thepreceding

ecircuitswhosetransientbehavioris dis-

nceFunctionsforSimpleCircuits

eoftheseriesRLcircuit,Eq.85,whichwe

ce

notation

sis usedasasymbolfor anycomplexfre-

omplexfrequencyofthesourceiss =jw,

eriesRLcircuitis Si=—R/L.Theresult

ationofthefrequencyfactorinthe impedance96through

torin thecomplexfrequencyplane.

sthatthe impedanceZisproportionaltothe

urcefrequencysandthenaturalfrequencysi

ultmayeasilybe visualizedwiththehelp

ninthecomplexfrequencyplanereferredto

Inthis splaneanycomplexvalueofs =a


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oint.Thehorizontalaxisor caxisisalso

xisor"axisofreals";the verticalaxisisthe

xis."Thecomplexfrequencyoftheun-

ecorrespondstosomepointonthej axis;the

Lcircuitisrepresentedby thepointSion the

fortheconstantmultiplierL,the impedance

chemanatesfromthepointSi andterminates

hejax is.

ghowtheimpedancechangeswithdifferent

ncyuof thesource,thediagramofFig.7 is

oneneedmerelyregardthepoints =juon

thisaxis.Thusthetip oftheZvectormoves

ngwwhilethebaseof thisvectorremains

variationofboththemagnitudeand theangle

earlyportrayed.

ofthe impedanceofthisseriesRLcircuit

he applieddisturbanceorsourceandthe

erelationbetweenthecomplexcurrentand

ntermsofthe

m'slawexpression

hichquantity,

whichjp

ppreciatethis

usrederive

entandvoltage

rthatdoesnot

- ......,..t f

reitheroneof these

nforr.J

.quantitiesasbeingtheexcitationorthe

regardthe

dLas aunitacrosswhichthevoltagee(t) ap-

rrenti(t)exists.Whatwewishto establishis

orthisunit,as thoughitwereasingle element.

ourse,is againthedifferentialequation

eareinterestedonlyinthe steadystatefor

unctionswewrite


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eckthepreviousones;yethereit ismanifestly

with103 representsthevolt-ampererelation

noimplicationastowhichquantity,the

e excitation.

alledadmittanceasthereciprocalofthe

fortheseriesRLunit maybewritteninsym-

tuallyinversepair

103and105.

tionsismoreconvenienttousewhen/ is

o befound,thesecondwhenEis givenand

ons106alsosuggest thatonemayregard

tageper unitofappliedcurrent,andthead-

urrentperunit appliedvoltage.Thus,in

mayberegardedas thevoltagevectorEwhen

Lamperes.IfIdiffersfromthis valueby

thenEdiffersfrom (s— Si)bythesame

rvedto havepreciselytheformofvolt-

esistancenetworks,withthedifferencethat

sistanceR andYinthe placeofconductance


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ct,oneispermittedtoconcludethatthe

moreextensivecombinationsofcircuitelements

theimpedancesoradmittancesoftheelements

einpreciselythe samemannerthatresistance

tsaredealtwithwhennumeroussuchelements

ousways.

entiallyreducesthestudyofsinusoidal

aviortod-ccircuitmanipulations(exceptfor

hecomplexquantitiesastime functions),

nofthecomplexvolt-ampererelations^ojrthe

ts.Thusfortheresistance

ethe complexvolt-ampererelationsforthe

ndcapacitanceelements,respectively.Thatis

onetheimped-

tanceYisG;

ttheimpedance

eYisl/Ls;

ttheimped-

mittance

nnectedin

anceisthe

heseparate

ectedinparallel,

sformedbyaddingthe admittancesofthe

pedanceconceptisa usefulonewhencom-

-

forthe


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ors(S— 0)and(s— Si),whiletheangle8of

0)whichis*r/2radians,minusthe angleof

R).It isclearthat,asu increasesfrom

creasesfromzeroand approachesunitytimes

artsfromir/2 andapproacheszero.

ereciprocalofZ,hasinfinite magnitude

radiansforw=0 andapproachesthemagni-

aswincreasescontinuously.

onin thecomplexsplaneisan important

s ofYorof Zforvariousvaluesofw. For

tedthephase

oltageandcur-

° or30° or

seeat once

construction

wouldhave

atR/Lwould

m-haveto beforaSiveQfrequencys= jw,

foretc.Itis alsoclearthatthecomplex

alfrequencySi=—R/Lcharac-

tatepicture

heseareusefulideas inthesolutionofanalysis

ke.

CcombinationofFig.11.Here

milartotheY andZofthe parallelRLcombina-

Eq.116.Theimplicationinvolvedherewill

n.Thegraphicalrepresentationinthe com-

similarto thatshowninFig.10 thatnosep-

cessary.


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f Fig.12itis readilyseenthat

elysimilartoZand Yobtainedfortheseries

mplexvolt-Fig.13. Pertinenttothecomplexvolt-

allelRCampererelationfor theseriesRLC

yEqs.103and 105;hencefurtherdetailed

d.

LCcircuitshowninFig. 13.Here


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sionoftransientbehavior,theimpedancemay

rosofthe second-degreepolynomialinEq.128;

— jwa

ecomplexnaturalfrequenciesoftheRLC

ofCh.5).Asdiscussedearlier,theimped-

actor—isa

ationfre-

naturalfre-

estion.All

is contention.

entgraph-

plane forthe

eofEq.129.

actors(s— Si),

givenbythree

onthepoint

ptforthefactor

tudeequals

of thevectors

dedbythe

6of the

of(s— s,)

s2)minust/2

nonemay

gesinmag-

gingpositions

edsiands2.

yinglocationsof Siands2for afixed

pectiongainedhereistremendouslyhelpful

,aswillbecomeabundantlyclearinthefol-

roblemworkassignedasexercises.

ation

rthe

esRLC


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ESONANCE

onance

noftheRLCcircuitit wastacitlyassumed

ientlysmallso thatthenaturalfrequencies

s,thenaturalbehaviorof thecircuitwas

Thistacitassumptionhasno influenceupon

Z,whichis correctlygivenforallvaluesof

129,inwhichthenatureofthe naturalfre-

easesRealaxisislocusof S]ands2

easesfromw0tooo

naturalfrequenciesoftheseriesRLCcircuitwithvariable

, (b)for«o<a<».

suponthe relativeamountofdampinginthe

e previouschapter.Arathercircumspect

nedthroughnoting,inthes plane,howthe

nds2changeforvaryingvaluesof a=R/2L

inFig.15.

tinenttoa valueswithintherange0< a

insto therangew0<a <oo.Fromtherela-

ctlytopart (a)ofFig.15 onereadilyrecog-

ndvaryinga,thes planelocusmustbea

=0,si ands2areat thepoints±ju0onthe

thecomplexnaturalfrequenciesmovealong

larpathstowardthecommonpointAonthe

fora =w0.Forthisconditionthetwo svalues

ehavioristhatdescribedearlieras critically

anfrequencyudhasjustbecomezero.The


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rjustfailstobeoscillatory,forits periodof

nite.

owthecomplexnaturalfrequenciesSiand

aincreasesbeyondthevalueu0. Itisobserved

nbecomedistinctbutbothremainreal,the

ereciprocalofthe magnitudeoftheother

2 =wo2forall

andSi —»— °c.

he oscillatory

bablyofprimary

inmanycommuni-

ationsthehighly

pedcircuitisof

ircuitsthebe-

atorycircuitis

lyforthereason

nableandhence

nsoughtafter.In

understandclearly

ehaviorof the

encethefollow-

ven.

ircuita sinus-

tamplitudeEis

s intheresulting

thelatter isequaltothe productofEand

s toberegardedasconstant,wemayas

hen/is numericallyequaltoY.

y

-

e interpretedwiththehelpofthe graphical

ationwhereais smallcomparedwithudand

alto w0.Theparticularappearanceassumed

forthisspecialsituationis picturedinFig.16.

tedinnotingthe behaviorofY(i.e./)with

nfrequencyo>ofthe appliedvoltagesource.

onofthisfigurethata rathercriticalbehavior

ighborhoodofthenaturalfrequencyoy«* wo,

)becomesrelativelyveryshortandthe

rge.Thus,asthevariablepoint s=ju


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ESONANCE

mplexnaturalfrequencySi,thelengthof

shortandthenlongeragain;correspondingly/

thenrelativelysmallagain.Thereexistsa

snearSiwherethe currentbehavesrather

iedfrequencys=jw,becomingexceptionally

eofbehavioriscalledresonance,andthe

mediatevicinity

esonancerange.

esonance

exactandcom-

onaswellasbeing

almeaningto

edpropertyof

tisonlynat-

nexceptionally

rivingfrequency

deswiththefre-

on.Ifdamping

nt(R=0),the

ponthej axis,

or(S— Si)inFig.

o—theresponse

yphysicalsys-

esenta sufficient

okeepthe points

eleftof the

eresponsewillalwaysremainfinite.

ands2arerather closetothejaxis(a «wo)

mFig.16 thatthroughouttheresonance

and(s— 0)remainessentiallyconstantand

alues2ju0andju0. TheexpressionforY,

givenverynearlyby theapproximateform

n

g.16

cal

eisto

eand

edviewof aportionofthes planenearthe

visualizingthebehavioroftheadmittance

Whenthe variablepointsonthe jaxisis

mittanceclearlyhasitsmaximumvalue


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ehereis zero,andsothecurrent isinphase

this point.Aplotof |Y| — |/| versusw

identlyis symmetricalaboutthepoint

18wherethe independentvariableischosen

splacementfromtheresonancefrequencya'0

usonesuchunitbelowand aboveuqcorre-

pointssa andS&inFig.17. Herethemagni-

wnto1/y/2=0.707of itsmaximumvalue;

resistanceR (proportionaltothesquareof

veofthe simpleRLCcircuit.

maximumvaluebyafactorof1/2.Hence

as thehalf-powerpointsontheresonance

pointstheangleofY andthereforeof/with

s,atthe lowerhalf-powerpointthecurrent/

,attheupperhalf-powerpointthe current

eenthehalf-powerpointsisusuallyregarded

sonancecurve,i.e.

requencyw0 tothebandwidthwi sameasure

Ccircuitwhenitis usedto"tune"tosome

sinthe firststagesofa radioreceiver.This

tyof thesimpleresonantcircuitis called

4onehasfor thisfactor

GofCh.5.


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OLARFORMSOFIMPEDANCEFUNCTIONS301

snotlarge,the representationofYasgiven

rcrude.However,theexactrepresentationin

nterpretationofits frequencyfactorsas

ficientlyclearandstraightforwardtoenable

yallrelevantfeaturesregardingthe de-

rameterss,Si, s2-Itis clear,forexample,

whensis directlyoppositesibutratherat a

ecauseof thelengtheningofthefactors

«.Also,theangleofYis notexactlyzero

allytheserefinementsareoflittleimportance,

thatthegraphicalinterpretationin Fig.14

venthesemoredetailedmatters.

rFormsofImpedanceand

AlternativeInterpretation

nceoradmittancesofar,we havebeen

xquantitiesintermsof theirmagnitudesand

havebeenconsideringthemintheirso-called

applicationsiti3moreconvenienttorepre-

orm",intermsof theirrealandimaginary

esistivepart"andX(w)the"reactivepart"

G(w)andB(w)are referredtorespectivelyas

the"susceptivepart"ofY.Thesetermsare

ngouttheword"part"and speakingsimplyof

ctance"ofanimpedance,orthe"conductance"

dmittance.Useoftheseabbreviatedtermsin

ofZandY canleadtomisunderstandingsince

d"conductance"arealreadybeingusedto

nductanceelement.Whilesuchanelement

evaluedependsonlyuponthe sizeofthat

u)and(?(w)in therepresentations136and

ctionsofacollectionofresistance,inductance,

s,besidesbeingdependentuponthesource

eimpedanceoftheparallelRCcircuitof

2


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Zdependsuponthevaluesofboth parameters

esourcefrequencyw.

fFig. 13wehaveaccordingtoEq.126

42)

mesidentifiedwiththeelementR alone,but

ervethat,ifweformthecorresponding

1/Co,)2

M=R2+ (Leo-1/C.)»(H4)

?(w)is notsimplyl/RbutdependsuponL,

fficetoshowthat,ifwer efertothereal

"resistance"insteadof"theresistivepart,"

kintheclarityof ourassertionunlessthe

carryingonthe discussionaresufficientlyex-

meant.Asimilarsituation existswithregard

artofan admittancesimplya"conductance."

reisany chanceofbeingmisunderstoodthe

ouldsufficetopreventconfusionwithresist-

asingleelementwhichwouldbedenotedas


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OLARFORMSOFIMPEDANCEFUNCTIONS303

onaldependenceuponwasimpliedin the

sed asadistinguishingmark.

tersXandB northenames"reactance"

edtodenotecircuitelements,sothatthesame

dindistinguishingthese quantitiesasthe

respectively.

eimpedanceoftheRLCcircuitas given

onvenientindiscussingtheresonancephenom-

erpretationofresonanceasaconditionfor

stantappliedvoltageis amaximum,itis

curswhenthereactivepartofZis zero;that

VC=w0(145)

one.Althoughthiswayofdealingwiththe

pleandchecksintheend resultswiththecon-

cksthedirectphysicalinterpretationof

ceornearcoincidenceofappliedandnatural

possessthesimpleanalyticinterpretationin

rmitsanapproximatestudyvalidinthe res-

method,moreover,lendsitselfwithout

omplicationstotheconsiderationofresonance

itswhichpossessseveralresonanceregions

aturalfrequencies.

r thestudenttounderstandeachimportant

ommorethanonestandpointifpossible.In

etationofresonanceasaconditionofzero

nterestinganduseful.Inorderto facilitate

edienttowritethe impedance141forthe

nductiveandthecapacitivecomponentsof

Xc.WithreferencetoEq.145,which

conditionintheseriesRLCcircuit,itshould be

cefrequencyisheregivenasthat oneforwhich

cancelsthecapacitivereactanceXc.The

nentreactancesarerespectivelypositiveand

aluesevidentlymakessuchcancelation


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EMENTARYIMPEDANCES305

he"width"of theresonancecurve(Fig.18).

to observethattheslopeofthe reactance

quencyw0isfoundfromEqs. 142and145tobe

teexpressionforXvalidnearw0 (equationof

=w0)wouldbe

gapproximaterelationfortheadmittanceY

a+ju0)]

= juandSi~ — a+ju0.

mentaryImpedancesandAdmittances

severaltimesinthese discussionsthat

arianttoaninterchangeofquantitiesin the

nce.

nce.

allelconnection.

uit.

evidentlyaddthepairofquantities

ce.

otheruponinterchangingvoltageandcurrent.

example,theimpedanceoftheseries RL

enticalwiththeadmittanceofthe parallel

Rvaluesarereciprocal(Rof oneequalsG

ysequalsCin farads.Thisresultisclear

—R/Loithefirstcircuitbecomesidentical

er,whencecomparisonofEqs.105and124

dancefunctionsshowsthattheZof one

eYoftheother.The networksofFigs.8

tionsbetweenelementvalues,aredualsof

dingimpedancesarereciprocals,andthecorre-

ereciprocals.Thustheproblemoffindinga


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pedanceisthereciprocalofthatofa given

constructingthedualofthat network.

rcuitofFig.9 andtheseriesRCcircuit

values arereciprocalandLinhenrysequals

ofEq.116 (pertainingtotheRLcircuitof

120 (pertainingtotheRCcircuitof Fig.11),

circuitofFig.13is givenbytheparallel

the implicationthatR'=1/R,U= C,

ension-

elatter

dentical

Un^*ance(seeEq.129)of

ersa.

phenom-

0we

mplex,,

eparallelshouldherecOnsiderthe'current*S

altothethesourceof constantamplitude

.andlook foramaximuminthevolt-

encyof

thenaturalfrequencyofthecircuitis varied

ncyconstant.InthecircuitofFig.20,reson-

ximumimpedanceorminimumadmittance,as

nthecircuitofFig.13 wherethereverseistrue.

edcharacterofresonanceinthe twosystems

dbyreferringtoit inthecircuitofFig.20 as

nthecircuitofFig.13 as"seriesresonance."

ago,whenengineersconsideredallsources

esponsesas currents.Thusamaximum

aysmeantaminimumimpedance.Incon-

umimpedance,asresultsinthe circuitof

henreferredto as"anftresonance."With

wayofregardingsourcesas capableofbeing

,useofthe term"antiresonance"isbecoming

sometimesemployedtodesignateacondi-

ncontrastto maximumresponse,whatever

citybetweenapairofimpedances,which

pertyandone thatissolvedin eitherdirec-


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EMENTARYIMPEDANCES307

theprincipleofduality.Anotherimportant

eenapairofimpedancesisthat inwhichone

entoftheother.Analyticallythecomplemen-

bythe relation

ntodiscusscompletelytheproblemof

these impedancesisfoundfromthatofthe

ssinterestingandusefultopointouthowa

-resistancenetworksthatinvolvereciprocalimpedances.

als.

worksmaybefoundthroughmakinguseof

p.Thusparts(a)and(b) ofFig.21showtwo

hichtheresultantimpedanceisapureresist-

arer eciprocal.Forthecircuitofpart(a)we

ance

etwoadmittancesadduptounity.Similarly

whichincidentallyisthedualofthecircuit

Z i1+Z1

heretwo impedances(thoseofZiinparallel

ithR)addup tounity.

uationin Fig.21(a)isgivenbyconsidering

8 inparallelwiththe seriesRCcircuitof

ethensimplyLsandl/Cs respectivelywhich,


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,are reciprocal.WithR=1,theadmit-

Lcircuitis

Ccircuit,Eq.120,becomes

kebecauseofthespecialchoiceofparameter

dentthattheadmittancesofEqs.159

ethattheseso-called"constant-resistance"

owninFig.21 forZ2=1/Zi,behaveatthe

iselyasaresistancewould.Thus,ifthe

mple,isexcitedby aunitvoltagestep,the

tepjustasit wouldbeforapure resistance

nsientcurrenttakenby eachofthetwo

elymustbesuchatimefunctionthat thesum

yunityforall t>0.In thecaseofthe

hesecurrentsis theexponentialbuildupina

onein theseriesRCcircuitisan exponential

anddecayrespectivelyareequal,withthe

twocurrentsequalstheconstantvalueunity.

xploitthedetailsof thisproblemaswellas

ercises.

mphasizinginconnectionwiththekind

opicof thepresentarticleisthe factthat

shipsbetweentheimpedancesZiandZ2ofa

pendentlyofthevalueof thesourcefrequency

mpedancesrelatedas Zi=1/Z2wehave

d Z2mayindividuallyvaryasfunctions

anner,yettheycontinuouslyandforevery

conditionZi=1/Z2.Thesameis trueof

onshipZi+Z* =1.Anotherimportant

discusslateron,namely,thatof equality

fferentconfiguration),willreceivethesame

hregardtoitsnondependenceupons.These

ausethereexistsanotherbasisuponwhich

tualrelationshipsof beingequal,reciprocal,

mely,whentheseconditionsareassumedful-

e only.Indealingwithsteady-statepower


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ENCYSCALING

hefixedfrequencyof60cyclesper second

estricteduseoftheterms "equivalentnet-

orks,"etc.Oneshouldnotconfusethese

withthevastlymoresubtle andinteresting

cyScaling

erelationsforthesingle-circuitelements,

sto befocuseduponthefactthat,for agiven

ltageislinearlyproportionaltothenumerical

aninductanceL,oranelastanceS= l/C.

ouldbeevidentthatif,in agivennetwork,

multipliedbythesamepositivereal constant

pliedunit currentismerelytomultiplyall

ppliedunitvoltagetomultiplyall currents

nleavestheimpedanceofa networkun-

yitbyA.This processisreferredtoas

oralsoas"changingtheimpedancelevelof

Sinceit caneasilybedoneatany stage

hoiceofanimpedancemultiplierisvery

processisthatpertainingtothe complex

simpleexamplestreatedabovethattheim-

nctioniscompletelydetermined(exceptfor

sseveralfrequencyfactorsofthe form

complexnaturalfrequenciesofthenetwork.

hatthissimplestructureof impedanceand

dsforallnetworks,themorecomplicatedones

uencyfactors.Inspeakingoftheimpedance

therthanofthephysicalnetworksthey

evarioussn valuesappearinginthefrequency

ciesratherthanasnaturalfrequencies.

animpedancefunctionmayberegarded

tionofdotswhenplottedin thecomplex

stellation,however,impliesachoiceof

fferentchoicesofthisso-called"frequency

tiondoesnotchangewithregardto thegen-

merelyexpandsorcontractsasitwould

dthesplanethrougha magnifyingglass,or

ectofsuch"scaling"upontheimpedance

ngor contractingthecomplexfrequency

zetheprocessthroughimaginingthe


/ h t t p : / / w w w . h





ber,whencescaling correspondstoauniform

nbothcoordinatesofthe plane.

the splane,incidentally,isreciprocal

fcomplexsvalueshavebeenseento be

maginarypartsareradianfrequencies.

atioandhence dimensionless,aquantity

condhas thedimensionofreciprocaltime.

rectionin thesplane,therefore,represents

mstretchofthes planeinbothdirections

sionofthetime scaleappropriatetotheprob-

rsa.

soradmittancesasfunctions ofthecom-

s importanttoavoidconfusioninone's

aluesandcorrespondingcomplexZvalues

n,ofcourse,alsoberepresentedgraphically

s thenreferredtoasthe Zplaneoras the

cyforthebeginnertoconfusethe splane

eY plane.Specifically,thescalingorstretch-

houldberecognizedas havingnothingtodo

butonlywithpointsin thesplaneat which

fwewere consideringplotsofthemagnitude

adianfrequencyu(donein theusualmanner

as ordinatesversusw'sasabscissae)then

lanecan bevisualizedthroughimagining

arubberblockwhich issubsequentlystretched

ssadirection.Ordinate valuesarethereby

ssaeatwhich theyoccurareinfluenced.

Z \versusgoor 0versuswis one-dimensional;

ensional.

ncyscalingisbroughtaboutthroughappro-

entvaluesof agivennetwork,oneneedmerely

heimpedancesof theseparateelements,

Thusthefrequencyshasno influence

sistanceelement,whereas,inan inductance

varianceoftheimpedancewithchangesin

andsvaryinversely.If inagivennetwork

ftunchangedbuttheL's andC'saredivided

enanyimpedancevaluethatpreviously

sanowoccursfor s=nsa:that is,atafre-

ence divisionofallL's andC'sbyn >1is

econstellationofcritical frequenciesinthe

e constantsntimessmallerandthe net-


/ h t t p : / / w w w . h





andtheimpedancemagnitude10,sothat

comes0.1.Ifwechooseavoltagescaleof

rentscaleof20 inchesperampere,thevectors

whereas,ifwechangetoacurrentscale of

vectorhashalfthelengthofthe Evector.

nsseveralvoltageorcurrentvectors,relative

aning,butrelativeangleshave.

me,wemaysaythata choiceofscalesfor

sascalefortheassociatedimpedanceZ

pearon thesame

alesforJ andZ

oiceofscalesforE

xample,achoiceof

peresperinchimplies

fthe scaledlength

inches)ofavoltagevectordividedbythe

aled{h (Jninches)Qf&vectofig

,,.,. ,....,

theappropriatelengthininchesfor

evector.Forthese

5incheslongrepresents25volts;a current

esents4amperes;thelengthof theassoci-

2.5/2.0=1.25inchesandrepresents5 X1.25

orZthescaleof 1ohmperinch,whenceit

and/becomeequal;thatis,the numberof

mber ofamperesperinch.Thistacitcon-

alwaysapply;andinfactit mayinmany

commodate.

gles,thatwespecificallyusetheterm

agramofFig.22(a) couldjustaswellbe

orinanyone ofaninfinitenumberofaddi-

entations.Theonesignificantfactwhich

ysisthat thecurrentlagsthevoltageby

onofthediagramas awholeisthus per-

to choosethatorientationwhichseemsto

calconditionsofthe problem.Forexample,

henitis customarytochoosetheangleofE

eisacurrent,theangle of/isusually taken

ese choicesthevectorEservesasphase

inthesecondchoicethevector/ becomes

hevervectorischosentohavezeroangleis


/ h t t p : / / w w w . h




cevector.Althoughseveraldifferentvoltages

vedina givenproblem,itisclearthat the

r ofonecurrentvectormayarbitrarilybeset

presentedinrectangularform,thevolt-

paratedintoasum oftermscorresponding

activecomponentsofZ,asin

61)

ponentsofErepresentedbythetermsIR(u)

stancedropandreactancedroprespectively—

responding

ne forthe

2(a),there-

owninFig.24.

(u)musthave

onasthevec-

apositivereal

actbystating

withthe

ponentgiven

d,clearlyis

thatis,it leadsthevector/by90°.This

ssedbystatingthatthe IXdropisin quad-

houghthis terminologyisabitambiguous

mpliesaright-anglerelationshipwithoutre-

andreactivecomponentsofEvectorially

softhesecomponentvectorsarefixed,for

6,assoonas alengthforthe vectorEischosen.

mmustcoincideindirection(mustbein

ts length(asalreadymentioned)isarbitrary.

blemsit maybeconvenientorusefulto

ectorIintocomponentsthatarerespectively

draturewithE;or itmaybeexpedientto

sofZintosubcomponents.Acommonexam-

earisesindealingwiththeseries RLCcircuit

.Here

re-

volt-

to


/ h t t p : / / w w w . h





Eq.147.

amforthis ex-

ochoosethecurrent as

standingthefactthat

rcefunction.There-

Fig.25,is drawnfor

pacitivereactance

thatthenetvoltageE

atterleadsthe voltage).

edropIX issmall

ponentIXlorIXc,so

E, whichincludes

smallerthan either

.IfXl+Xc=0, we

onforwhichE= IR

makesmoreevident

,onemayhavevolt-

eandcapacitanceele-

bemanytimeslarger

e.Forthis reason

nbeexercised

resonanceinthe

ersourceusedissmall

e possibilityof

cidentalcontactwith

tratesthecircum-

useofavectordiagram,isthe circuitsche-

whichconsistsofthethree impedances

ewe write

mas o— zx— z2—z3—o

entation

arbi-

rdiagramtraryimpcdanccsinseries,

osenas

mpedancesZiandZ2areassumedto bein-


/ h t t p : / / w w w . h




ANCEFUNCTIONS315

ndX2> 0),whileZ3isassumedto be

<0).Thediagramshowsallthreeimpedance

osstheseparateimpedances)brokendown

components,as

ichequals

mspection

srelativeto

ationshipsofall

ecommon

equalmeas-

crelationship

of thevector

e.

avebeen

ncetotheim-

kingEand/,

equallywell

alprocedurein

ameterY.

Fig.26 wereconnectedinparallel,such

ebasiswouldbeindicated.Thedetails

be exactlyanalogoustotheonesgiven

nd/interchanged,R'sreplacedbyG's,and

m

of

ce

ellas

evec-

nceFunctions;TheirPropertiesand

rt.2ofCh. 5,thedifferentialequation

partofanetworkwithvoltagee(l) atthe

rpointisalwaysofthe form

e(165)

m• .•b0are realconstants.Theyareallposi-

samepoint inthenetwork;otherwise

smaybenegativeaccordingtowhether

nction.*

aborateduponinArts. 4and5of Ch.9.


/ h t t p : / / w w w . h





me", theparticularintegralyieldingthe

sthavethesameform.Henceforthesteady-

ntialEq.165 itisappropriatetosubstitute

60)

ais+ a0)/e"=

•+b1S+ bo)Ee"(167)

onfactore'i,onehas

a1S+a0P(S)

biS+b0Q(s)

Q(s) arefactoredintermsoftheir zeros,

mestheform

m)

sitiverealconstant.

etransient(force-free)partofthesolution

th e=0.Assumingfor thesolutionto

ntialequationtheexpression

tutionto

on(Aj£0) demands

•••+aiP+do=0(172)

quationdeterminingthecomplexnatural

ththetransientcurrent.Weobservethat

nciesSi,s3,• ••s2n—1appearinginthe numera-

Thecompleteresponse(transientplussteady

A2n^e'^1+— e"(173)

transient(force-free)partofthesolution

th i=0.Assumingfor thesolutiontothis


/ h t t p : / / w w w . h




ANCEFUNCTIONS

lequationtheexpression

^ 0)demands

+•••+6iP+b0=0

quationdeterminingthecomplexnatural

ththetransientvoltage.Weobservethat

nciess2,S4,. .•,s2mappearingin thedenom-

69.Thecompleteresponse(transientplus

egivenby

Ai•• •A2n-iinEq.173 andBi.• .B2m

dfromtheknownstateofthenetworkatthe

edandthe demandsmadebythesteady-

thatsameinstant,the discrepanciesbetween

equantitiesuponwhichthe sizesofthese

etailsoftheirdeterminationdonotinterest

nificantto pointouthoweverthat,fora

sults173 and176aremeaningful,evenwhen

unctionsarezero.Whene(<)andi(t)refer

etworkandthevoltageexcitationiszero

representtheshort-circuittransientbehavior

efora zerocurrentexcitation(/=0) Eq.176

en-circuittransientbehavior.

s(si.• •s2n-i)andpoles(s2 •••s2m) ofthe

reidentifiablerespectivelywiththeshort-

mplexnaturalfrequenciesofthepertinent

impedanceforany linearpassivelumped

cated,hasthesameformin termsofitsfre-

dthesamephysicalsignificanceregarding

volvedinthesefactorsasdoesthe impedance

andthree-elementcircuitsdiscussedinthe

thofthisstatementandsome ofitsuseful

eadditionallyplausiblethroughconsideration

•+ B*J»* +IZ(s)e'

,seethediscussionin Art.4ofCh.9.


/ h t t p : / / w w w . h





sthe logicalextensionofthesimpletuned

or ofitsdualas describedinArt.8andillus-

ble-tuned"circuitwhichiscommonlyusedinradio

roniccircuits.

lythecircuitwewish tostudyconsistsof

upledbymeansofa smallcapacitance,as

roblembe

pedance

essionforthis

roughuseof

s,suppose

ohalves

cuitvoltage

s showninFig.29.Thispart ofthecircuit

discussionsinthischapter.DenotingbyZ

llelRLCcombination,weobtain

alysis

tin

gtothepreviousanalysisof thissimple

ent (Fig.28)isgivenby thequotientof

ceencounteredbythiscurrent,whichis

putvoltageE2 isrelatedto/„ inexactly


/ h t t p : / / w w w . h




ANCEFUNCTIONS319

s relatedtoI\(Fig.29). Hencewehave

178 yields

2+2as+wa2+ (2Ci/C)«2)

tionofthisresultbecomesinterestingif

obesmallcomparedwithC (oftheorderof

g.29we willassumetobehighlyoscillatory

quadraticfactorin thedenominatorof

ten

(1 +2C1/C)(s2+20s+wb2) (184)

elyreplaceableby

tionsindicatedinEqs.185,186,and187,

t,if aquantityxis smallcomparedtounity,

rly1— x,andVI+ 2xisverynearly

oughmakingthe appropriateseriesexpan-

areand higherterms.Inarrivingat the

84,wefirst ofallregardthefactor

qualto unity,andthennotethatthe dif-

s negligiblecomparedwiththedifferencebe-

tselfissmallcomparedwithw0.For example,

da is1percentofw a, thenw 0— w 6isroughly

/3isonlyaboutl/50th of1percent ofwa.

sresult istosaythat, ifCi/Cisextremely


/ h t t p : / / w w w . h





draticfactorinthedenominatorofEq. 1S2

thefirst,but, asthevalueofCi/Cbecomes

ctofitspresenceisnoticeablein thediffer-

jsdetectableinthedifferencea— 0inthe

ntorobservation.

es,whicharedescribedbysayingthatthe

combinationsinFig.28arelooselycoupled

Ci,wefindforthetransferimpedance(Eq.

,!2)(s2+2«sW)

nby Eqs.179and187.

(s-s2)

— ua

)(s-S4)

— jub

eralformgivenin Eq.169,namely,

)

neof thecriticalfrequenciesofthisimped-

g.30. ThevaluesSi,s2,s3,s4t whichare

areindicatedbycrosses;thedifference

eredtobeof thesameorderofmagnitudeas

allcomparedwitheitheruaor wj.Thethree

meratorofZ12allhave theform(s— 0).

icatethecomplexfrequenciesatwhichthe

his reasonaresaidtorepresentthe zerosof

Zi2(s)hasthreecoincidentzeros(also called

.Zerosareindicatedin thesplaneby small

orderzero ofZi2ats =0is indicatedby

clesatthispoint.

againregardedas apointthatmoves

ntpassesalongsideofthe pairofpolesSi

behaviorsimilarto thatobtainedforthe

Ccircuitunderresonanceconditions,with


/ h t t p : / / w w w . h




ANCEFUNCTIONS321

hattheeffectisnowenhanced,owingto the

the resonanceregioninsteadofonlyone.

edviewof thefrequencyfactors(s— Si)

anceregion.Heretheotherfactorsin the

thecomplexFig.31.An enlargedportionofthe

es(crosses)and s-planepictureinFig.30 pertinentto

ferimpedancethediscussionoftheresponsecharacter-

-tunedcircuit isticsshowninFig.32.

substantiallyconstant(i.e.,s— s2*** S~~s4

t anapproximaterelationvalidforthe

es

equencysinceit referstothecenterof the

eristicsonthepart ofthefunctionZ12(s)

herelativevalues of(wa— «6)=2aand a.

relativelyfarapartcomparedwitha,it is

1 thatthemagnitudeofZi2as afunction

ma,oneatthepoints =juapproximately

danotheroneoppositeSi.Our presentcircuit

frequencies.


/ h t t p : / / w w w . h





es (2ainFig.31) relativetoais decreased,

veof Zi2vs.umovetowardeachother.At

e double-humpedcharacterofthiscurve

antbehaviorissimilarto thatshownearlier

Ccircuit,exceptthattheresonancecurve

s onepassesbeyondtheresonanceregion.

secharacteristics(knownasdouble-humpedresonance

edcircuitofFig.28.Thecurveof (a)isforthe so-called

entcorrespondingtocoincidenceofthemaximashownin

=wo.Thedashedcurvein(a) isthesimpleresonance

ddedforcomparisonpurposes.

tdoublyresonantcircuitexhibitsahigher

hesimpleRLCcircuit.

Fig. 32wherepart(a)is drawnforthe

art(b)a= 2a.Inthelatter casethecurve

character.Theconditiona=a,forwhich

,correspondstothecriticalcasein which

ewiththe minimumatu=u0.For asmaller

ssingle-humpedin character,butthemax-

dropsbelowthevalueR/2.

coupledcaseshowninFig.32(a),wehave

esignificanceofthe parameteraasshown


/ h t t p : / / w w w . h




ANCEFUNCTIONS323

(Ci/C)«o=a (195)

egreeofcapacitivecoupling.Equation193

en

ofthe frequencyfactors(s— Sii)and(s— s3)

s evidentbyinspectionofFig.31,and,

tcapableofrealizingthe transfercharacteristicsofthe

.28aswellas othermoregeneralones.

we seethatthemidbandvalueofZi2

.32.Thedottedcurveinpart (a)ofthis

ofthesimple resonantRLCcircuitandis

comparison.

circuitsthatcanbe devisedtohavethe

aracteristicasthisso-calleddouble-tuned

erobviouswayof achievingasimilarresult

pleresonantRLCcircuitsappropriatelycas-

tha single-tunedcircuitlikethatinFig.29

tofthisimpedanceinserieswith itsinput

putimpedanceequalstheresistanceRand

aceofRin anothersingle-tunedcircuit.The

atedasoftenas weplease.

sbasisis showninFig.33.Theparallel

ohmat theright-handendformsthefirst

edance


/ h t t p : / / w w w . h





ertinentcomplexnaturalfrequencies.The

f1ohmin parallelwithaseriesLCcombina-

he firsttunedcircuit.Inthis complement,

ysandthe capacitancehasLifarads,as

21(b)andpertinenttext,notingthatthe

hedualof theparallelLiCicombination].

oftheterminalsa-a'is thusequivalenttoa

takestheplaceof a1-ohmresistanceinthe

vingL2andC2,forwhichthe impedanceis

exnaturalfrequenciesappropriatetothis

ngtotherightat theterminalpaira-a'

hevoltageata-a'andthe currentatthispoint

ltoIiZ2,whencetheover-allresponsefunc-

mpletecircuitofFig.33 canbemadeto3neld

enby thedouble-tunedcircuitofFig.28.

tquiteaseconomicalfromthestandpoint

butit affordsgreaterflexibilityinthevariety

able.Thusthepolesin Fig.30canbemoved

maintainingpairsofconjugatesofcourse)

henthe networkofFig.33isused,while the

withthenetworkofFig.28are notonly

mplyrelatedtotheelementvalues.Thusone

sin Fig.30coincideinpairsso astoobtain

hatisessentiallythesimpleresonancecurve

g. 28thisconditioncorrespondstoCi= 0;

omecompletelydecoupled,andtheresponse

ofFig.33,however,thisspecialcondition

anyother.

yvariationstothethemesuggestedby the

an useseriestunedcircuitsinsteadof

ecanuse thesetwotypesincombination,

weuse aseriestunedcircuit,thenits com-

parallelsinceour reasoningthenisdonein

eadofimpedances.Althoughitisnotour


/ h t t p : / / w w w . h




scussthesematters indetail,thereadercan

beensaidthatimpedancefunctionswithany

calfrequenciesinthesplane areconstructive

omeof theserequirefortheirunderstanding

dthanthesimplefundamentalprinciplesset

circumspectiongainedthroughrepre-

quotientof frequencyfactors(Eq.169)and

neasameansfori nterpretingthebehavior

ctorsasfunctionsofthevariables =ju.

yvisualizethepropertiesofdouble-tuned

es thebehaviorofthesingle-tunedcircuit.

ergeneralizationtocircuitspossessingany

nceismadeevident.Thesemattersimpress

creasingclarityasweproceedwiththe con-

topicsinnetworktheory.

samescale.

he effectsofintegrationuponasinusoidalwave?

ethe effectsofdifferentiationuponasinusoidalwave?

ntheanalyticalexpressionforthiswaveandcheck

n.

Commentontheresult.

quarewaveoftheform shownaboveis

-c os51)

msandobtaintheirsum graphically,(b)Supposingthat


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e




edadistanceof7r/2to theleft,writedownthenewFourier

umbers

|.

howfromageometricalconstructionthat| B+A |

umbers

R e[A]± R e[B)andIm[4± B)- Im[A]± Im[B].

e[A]R e[B]- Im[A]Im(B)andIm[AB\ = R e[A]

e. Re[AB]* Re[A]Re[BJ.

sreal. WhatdoesthisresultbecomewhenA =

0.2)].

dance

he polarformZ=| Z\/e.(b)Expressit intheform

/ n.

b amperesinanimpedanceZ- R+jX ohms,find

irectionofthecurrent.Expressthe resultin(a)rectangu-

exponentialform,(d)asaninstantaneouscosinefunction

(e)What isthemagnitudeofthevoltage?(f) Whatis

hecurrent?

mplexnumberstopolarform;that is,findthevalues

tformAe'6.Express6 indegreesorradians,whicheveris

umnabovecanbedonewithoutslide ruleortables;try

ddition inthefollowingexamples.Statetheanswer

larform.Theycanbedoneby inspectionofpertinent


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



0°

0°

07/45° -0.707/-45°

.7/-105°

7/-105°

sionsintoas compactaformaspossible,usingthe

.Includewithyoursolutionsthepertinentvectordia-

c os(w<+60°)

n(wt-30°)-sin(wt+60°)

^200i+0+2cos ^300* +^

00* -0+V %sin(300i-0

expressionsgivingtheanswersbothin rectangularand

)

7)'

asteady-statea-ccircuitis givenbytheformula

e,Z=compleximpedance.Fillinthemissing values


/ h t t p : / / w w w . h





laurinexpansionsfromthelistgivenbelow,evaluate

complexexpressions,retainingonlyfirst-orderterms.Es-

uranswers.

j

O.lj)

fTV— logTV/loge =2.3027log.VwherelogAT =

+jv ==Ae*,and,fromthe relationsA-

y/x),deducethatthepolaramplitudeA asafunction

xorof2/ isgivenrespectivelyby

,9)=— sin0/y

polarlociofsin 8andcos6, establishthatthelociof

dvariableyor forvariablexandconstanty arethose

tches.Fromtheseresults,findthe correspondinglocifor

taasis doneinthesketchesbelow,andshowthatthe

lieswithintheunitcircle abouttheorigininthe (u+jt)

sfunctionfor x>0 occurfrequentlyinelectricalas

blems.Forexample,theyformthebasisfor auseful

hartknownasthe"Smithchart.")


/ h t t p : / / w w w . h




sedtoproducerangemarkerpipson theindicatorscope

acuumtubeisconducting,a currentt'z,flowsthroughthe

negativepulseisappliedto thegridofthe vacuum

eonthegrid cutsthetubeoff,and thecurrentthrough

e circuitinthefigure willthereforeoscillateasanordi-

alcurrentt£ throughtheinductor.Theseoscillations

ng(clipping)circuitwheretheyareformedintoalmost


/ h t t p : / / w w w . h





passingthroughadifferentiatingnetworkthepulsesap-

tepositiveandnegativespikes,equallyspacedalongthe

pendingonlyon thefrequencyofoscillationoftheLC

esareremovedandthe positivespikesappliedtothe

idistantmarkersappearon thescreen,andthedistance

dicationonthe scopecanthusbedeterminedwithrefer-

removedfromthe gridattimet =S,thetubestarts

wdampingresistanceacrosstheoscillatingLCcircuit.

e,rapidlydieout.Bychoosingpropervaluesforthe param-

ketheoscillatingcircuitcriticallydampedwhenthetube

donthefigure.

markersatintervalscorrespondingto2000yards,cor-

oscillationof12.2microseconds.IfL=15millihenrys

producethedesiredresult?Theinitialcurrentthrough

atis theamplitudeofe*?

lenttuberesistancebe inordertomakethe oscillating

henthetubeisconducting(aftert= 5)?

sasshownin thelowerdiagram,andifour input

negativepulseec asbefore,isanegativestep ei(t)=

eintervalSduringwhichthe tubewillremaincutoffif we

sconductingabruptlywhenec= —125volts?LetRt=

micromicrofarads.Thegrid-to-cathoderesistanceofthe

nite.

Ii,I*,7jin theaccompanyingsketchrepresentthe

usoidaltimefunctionsofvoltageandcurrent.Specifically,

nby

es:R= 1ohm,L=1 henry,C=1farad,and u=1

complexamplitudesIi,h,/j, /,andtheassociatedex-

eouscurrentsii(<),*2(0,*j(0.*(').Plot avectordiagram,

tudesofvoltageandcurrent.

madein theanswerstopart(a) ifthevalueof Lis

madein theanswerstopart(a) ifthevalueofC is

tanceandcapacitancevaluesarechangedtomicro-

multipliedby10-6)whiletheradianfrequencyis changed

cond,howaretheanswersaffected?Howaretheyaffected

to110 volts?


/ h t t p : / / w w w . h




=cosI.Find Iandiff}■=Re [/«*"].

e givenquantityisi(<)— cost.(a)Find Eande(<).

nctionoftime inthe1.5-ohmresistor;inthe 0.75-farad

wersto part(b)ifthe givenquantityise(t)= cost,and

computationalprocedureinthetwocases.*

5,showthat,if wewrite

rR =1ohm,andthe range—ir/2<9< ir/2at15°

the tipsofthesevectors.Showthat theresultsapply

withtheexpressionsfor YandZinterchanged.

t]=cos<.Find:

ingthecomplexquantitiesE, I,E\,Ei,

quantityis i(t)=cost, finde(t),«i(<),eiQ),andplot

allthe complexquantities/,E,Ei,Ei.


/ h t t p : / / w w w . h





,henrys,farads

]-cost.Find

ctordiagramshowingE,,Ei,Et, I.

d*i(0,i(<),e(0,ec(0,andplot thevectordiagram

plitudesofallfivetimefunctions.

.Findi(<)ande(0.

,henrys,farads


/ h t t p : / / w w w . h




ndNote.In Probs.22through25Thevenin's

beusedto goodadvantage.

withR=1 ohmandL=1 henry,theimpedanceis

ittanceY(s)— l/(s+1),the latterbeingnumerically

urrentpervoltapplied.For certaintypesofanalysisit

ci)intoits realandimaginarycomponentsandstudy

nof theradianfrequencyu.Computeandplotthese

ange0< u<6.Nowshow thattheseplotsdonot

larnumericalchoiceR=L =1andto thefrequency

e usedtorepresenttherealand imaginarypartsofthe

veranappropriatefrequencyrange.

tfora seriesRCcircuitwithR= 1ohmandC — 1

maginaryparts ofY(ju)againovertherange 0<u< 5,

mpletegeneralityoftheseplots.

lcurvesofProb. 26apply(throughuseofthe principle

maginarypartsoftheimpedanceZ(ju)ofa parallelRC

C=•1 farad;and,throughaprocessofgeneralization

Probs.26and27,showthatthe resultsapplytoanyparallel

lcurvesofProb. 27apply(throughuseofthe principle

maginarypartsoftheimpedanceZ(ju)ofa parallelRL

L=1 henry;and,throughaprocessofgeneralization

Prob.27,showthattheresultsapply toanyparallelRL

n Probs.26through29applyin essentiallyunaltered

angle(that is,tothepolar representations)ofY(jw)and

thetwo relevantsetaofuniversalcurves,anddiscuss

entspecificcircuits,parametervalues,andfrequency

cuitin parallelwithaseriesRCcircuitso astoform

eY(ju)is,of course,givenbythesumofthe twoadmit-

LandRC circuitsrespectively.WiththeplotsofProbs.

easonablygoodsketchesofthem),considerhowyouwould

mhos,independentofthefrequencya.Checkyour

isexcitedbyaunit stepvoltage.Iftherelativeparam-

jo>)= 1/R,determineandplotthetransientcurrents

rallelbranchesof thiscircuit,andshowthattheirsum


/ h t t p : / / w w w . h





mperes.Doesthislast resultagreewithwhatyouex-

hy?

ob.31, makinguseinthiscase oftheuniversalcurves

thepertinentcircuitarrangement,andworkoutall

ships.

edualofProb.32,drawingthe circuit,specifyingthe

blemanditssolution.

yoccursthata seriesRLcircuit(excitedbyadirect

stsuddenlybeinterrupted.Thecontactsinvolvedinthe

eakerin asituationofthiskind mayrapidlydeteriorate

petitionsofcircuitinterruption,particularlyifthe excita-

inwhythis isso,and,in thelightofthe resultsof

yinwhichthe situationmightberelieved,assumingthat

feasibilitycan

hod,incidentally,

allytorelieve

somed-ccontrol

etchshowsa

sionofthe pre-

henR"=L/C.

osedforalong

ditionobtains.

tehis suddenly

rrenti(t)inthe remainingclosedloopaswellasthe volt-

tancefort>0. Whatisthevoltageacrosstheparallel

circuitsfort >0?Whatis thevalueoftotalstored

0,and wheredoesitreside?Describequalitativelyhow

ws aftert=0.

ove,R* =L/C.TheswitchSclosesat t=0.For

gesez,(f),ec(t),aswellasthe currentsii(<)andictt)cir-

ftera steadyconditionobtains,theswitchSissud-

etwovoltagesandthe twocurrentsasfunctionsoftime

rcuitinterruption?

isappliedto aseriesRLcircuit.Theresultingvector

erelaggingatanangleof 45°.Iftheradianfrequencya

esofRinohms andLinhenrys?Ditto fora=2x X60,

tallthis forE=100 volts.Nowrepeatallofthe above

0°;againfor60°. IftheappliedVoltageasa timefunc-


/ h t t p : / / w w w . h




sut(with Ereal),whatarethe expressionsforthecurrent

ateto thevariouscasesspecifiedabove?

ithIreal)is appliedtoaparallelRCcircuit. If

oltageistobe e(t)=62.4sin(U +30°),whatarethe

hmsand Cinfaradswhen(a) a=377radians persec-

persecond,(c)u =5000radianspersecond,(d) a•=109

re,E=1 volt(referencephase)andthecurrentsIi,

itude.Theirphaseanglesare respectively<)>i=0°,

ct>i =60°.Youareto findappropriatecontentsforthe

ot exceedingtwo-elementcombinationsincomplexity.

econdtostart with,andlaterconvertyourdesignto the

dagaintou =2rX 1000.Ifthereexistothersolutions

tatewhattheyare.Drawa vectordiagramshowing

ltantcurrentIq.Obtain anexactanalyticexpression

anglesinvolvedare

atchangesin thecir-

he circuitsshown

chesareinohms and

pedanceZineach

omplexfrequencyS,

formofaquotient

easketchofthe s

frequenciesandsome

Byinspectionof

\and6(the angleof

twhatuva lue

slargestvalue?

u. If aunitstepcur-

uit,findtheresultant

utterminals.If the

dsothat thepoint

0(andallotherpointsarechangedin thesameratio),what

esbecomein ohmsandhenrys?Whatisthe effectupon

Whatistheeffectupon thetransientresponseob-

he circuitsshowninthesketchesare inohmsand

pressionfortheimpedanceZ(s)in eachcase,andputit

quotientoffrequencyfactors.Plotthecritical frequencies

toeachZ.Do theresultssuggestanythingofinterest


/ h t t p : / / w w w . h





?Ifthe frequencyscaleisstretchedsothatu

he elementvaluesbecome?

dualto thosegiveninProb.41;that is,onesthatwill

faunit voltagestepisappliedtoeither oneofthese,

entcurrent(using theresultsfoundinProb. 41,ofcourse)?

escorresponding

quencyscalethatshifts

aredual to

t is,thosethat

es.Writetheir

frequency

ementvaluesin

dingtoastretch

tshiftsu — 1to

alfrequencies

41 findnet-

respectivelyin

workwillyield

ualtounityat

y, find(bythe

xt)thosenet-

taryimpedances.

rksbeusedinterchangeably

ceresultants?

complementarytothosegivenin Prob.42.

he networksshownattheleftarein ohms,henrys,and


/ h t t p : / / w w w . h




nsforZ\(s)andZi(s) asquotientsoffrequencyfactors.

chthelocationsofitscriticalfrequenciesinthes plane.

se impedancesrelated?Computetherealandimagi-

nces,andsketch(neatly)versus«for therange0<w <3,

onesheetandbothimaginarypartson another.

b.47 determinethetransferimpedancesZn=Et/I\

frequencyfactors,andsketchthecriticalfrequenciesin

quaredmagnitudeofZyi(jw)in eachcase,andsketch

t forrange0< w<3.Comparewiththereal-part plots

networksgiveninProb.47,andcombinetheseso as

ncecombination.UsingtheresultsofProb.48,whatare

Kb=Ii/Eifor thedualnetworks,andwhataretheir

nctionsofu?Doesthe constant-resistancecombination

ication?Howwouldyourevisethisresultantnetwork

ofthefrequencyscalethatputs thepointu= 1atw =

riesRLCcircuithas theform

.1— jl.Ifyoudrewthe resonancecurveforthis

esonancefrequencyandthewidthofthe curveatits

theQof thecircuit?Whatarethevalues ofitsparam-

farads?Howdo theseparametervalueschangeifthe

edbyafactor10,000(soas tomaketheresonancefre-

er)?Howdothecriticalfrequencieschange?Howdoesthe

veatthehalf-powerpointschange,andwhatis theeffect

etervalueschangeif Yisto become1000timeslarger

hischangehaveanyeffectupon theshapeofthereso-

turningto theoriginalsituation,supposetherealparts

ciesarechangedfrom—0.1to —0.01,whatare(a)the

thewidthoftheresonance

nts,(c)theQ ofthecir-

ues?

thesketch,showthat

theexpression

frequenciessi,«2,«sin termsProb.51.

G. Ifsi= -0.1+/10;

.1,whatare thevaluesofR,L,andG relativetoC?If

urrentsourceandaresonancecurveis takenforthevolt-

cefrequencyandthewidthat thehalf-powerpoints?

hatisthe magnitudeoftheimpedance5percentabove

vetoitsvalueat resonance?SupposethevaluesofRand

,keepingthequantity(ft'/L)+(G'/O -(R/L)+(G/C),

ntheimpedanceoradmittance?IftheQof thecircuitis

nificantsofarastheresonancebehavioris concerned?


/ h t t p : / / w w w . h





wnin Prob.51butwithG =0is tobedesignedto

maximumabsolutevalueof100,000ohmsat afrequency

ond.Atfrequencies10 percentaboveandbelowreso-

gnitudeshouldbenotmorethanone-tenthof 1percent

atarethe appropriateparametervalues?Whatisthe

hedataarechangedby requiringthattheimpedance

allerthan1 percentofits resonancevalueat10percent

,whatthenaretheanswersto theabovequestions?

hecurrentandvoltagesourcesare

0sin 5t

etimefunctionse(t)and

etworkNat theterminals1-1'hasthe form

enciessi,«2,S3,arelocatedin thecomplexfrequencyplane

panyingsketch.Iftheappliedvoltageisgivenby


/ h t t p : / / w w w . h




nallrealpartsofthe criticalfrequencies,computation

lesofthe factors(s— s»)toanaccuracyof afewpercent

dthatyouformulate,bymeansofasketch,a pictureof

eforemakingtheactual computations.

ofwhichrepresentsthesteady-stateperformanceofa

hetypethatsuppliesabout95per centoftheworld'selec-

diagram.e,(t)istheinternal generatedvoltage,c£,(<)is

Lis knownasthesynchronousreactance.If

+<p),\<p| <90°

ncy=/— 60cyclespersecond

ctordiagramand byuseofcomplexalgebra.Isthere


/ h t t p : / / w w w . h




ate

ements

acitanceCarefrequentlyreferredtoasthe

ergyisstoredin theirassociatedmagneticand

thisenergyis lostwithintheseelements

on,theyarenotcontaminatedwithresist-

element.Anyenergythatmay,throughout

sorbedby aninductanceorcapacitance

ewhollyreturned.Therateofenergy

erasufficientlylongtimeinterval,mustbe

gepowertakenbysuchanelementin thesinus-

zero.Let uscheckthissituationmore

energyinthe inductanceandcapacitance

studentsfromtheirfundamentalstudiesin

T=\Li2 (1)

=\Ce* (2)

espectivelythecurrentintheinductanceand

acitance.Thesemaybeanyfunctionsof

Varecorrespondinglytimefunctionsthat

nergiesat anyinstant.Theyarespokenof

taneousstoredenergies.

pererelationforthe inductanceorcapaci-

ns1and2maybe writteninalternativeforms

ubsequentdiscussions.Thus,fortheinduct-

amiliarrelation


/ h t t p : / / w w w . h




ELEMENTS341

themorebasic one

fluxlinkagespertinenttotheinductiveele-

ds

written

ywritein placeof1

r reciprocalinductance(nonamehasas

toredenergy

nmatches1 or2.

ceelementonehas

rrentisthecharge.If wewriteforcharge

pacitanceis elastance(denotedbythe

xpression2forstoredelectric energymaybe

andcurrent;and^and qarefluxlinkageand

meintegralsofvoltageandcurrent).


/ h t t p : / / w w w . h




NTHESINUSOIDAL STEADYSTATE

ementsWhenVoltageandCurrent

aninductance_Ljs^smusoidal;thatis,

17)

aneousstoredmagneticenergy

8)

inFig.1.Wesee thatthestoredenergy

enzeroandamaximumvaluewhichis \L\I|2,

nergystoredinthe magneticfieldassociatedwithan

ntinit isasinusoidof radianfrequencya.

gtwicethatof thecurrenti(t).Alternatively,

nctionoftime consistsofaconstantcom-

ternatingcomponent.Sincetheaverage

theconstantcomponentistheaveragevalue

odesignated.

uctiveelementis givenbythetimede-

onehas

equencysinusoid,.Itsaveragevalueis

gparagraphoftheprecedingarticlewhereit


/ h t t p : / / w w w . h




ATIONSINACOMPLETECIRCUIT343

geelementcannotabsorbenergyindefinitely

wouldultimatelyhavetoburst).

eethatthe peakvalueofthestoredmag-

cethe averagevalue;thatis,Tpcak=2Tav.

ot necessarilytrueforacircuit containing

soresistancesandcapacitances.Insuch

oneusuallyfindsthatTpcay< 2T,flV)orthat

torycomponentofT issmallerthanTav.

argerthanTavsincein thateventTwould

omeportionsofacycle,a resultthatmani-

ssiveelements.

ecapacitiveelement.If inEq.15welet

sameappearanceasthat ofT.Again

cuitswithR'sand L'saswellas C'sthat

signholding onlyinspecialcases.

the energyrelationsinaseriesRLC circuit.

onsina CompleteCircuit

riesRLCcircuitof Fig.2excitedbythe

miliarequilibriumequationreads

h.10.


/ h t t p : / / w w w . h




INTHESINUSOIDALSTEADYSTATE

equationis multipliedbyi(t).Wethenhave

(28)

O

q.28.Thethird term,throughuseof12,

7isthus seentobeequivalentto

2)

e conservationofenergy.ThusRi2repre-

teofenergydissipationbythe resistiveele-

V)is theinstantaneousrateofenergy

elements;ande,X tisthe instantaneous

the source.

cularformEq.32takeswhene,(t)and

dout inCh.6we mustwritehere

alpart"operatorand writingjustthesingle

equadraticoperationsareinvolved.The

e, however,thesameasbeforeandare

mpedanceintheusualmanner.The bar

e.

ingto12,we havethroughformallyinte-

)


/ h t t p : / / w w w . h





orchargeoneobtainsina straightforward

H(36)

2/7)(37)

ewritten

(38)

storedenergyfunctionsaccordingto 14and

40)

1)

pressionsarerespectivelyTav(Eq.19)and

givenin Eq.24sincethe capacitancecur-

ancevoltage|E |multipliedbyCos).In

esofTand Vtheseconstanttermsdrop out,

tsintoEq. 32gives

+ — J/ V2" '= e, X

deisconcerned,Eqs.33 and34yield

Jtf2*1](45)

deofEq.44,one shouldnotethat

uitimpedance,andthis multipliedby

E,.Hence Eq.44canbe written

^](46)


/ h t t p : / / w w w . h





stantaneouspowersuppliedtothecircuit

ations45and 46showthatthispoweras a

a constantcomponentwhichis

watts)

e-frequencyalternatingcomponentgivenby

uppliedpoweris evidentlygivenbythecon-

ereforeappropriatelydesignatedasP„.

ncevector(whichimpliesno lossingen-

same.TheangleofE,Ior theangleofE,I

of theimpedanceangle6,sothat

49)

50)

his resultisshowninFig. 3wherepart

duct\ E, \ X| / | cos8beregardedas| E, \

absorbedisexpressibleeitheras (1/2)theproductofthe

omponentofthecurrent (a),oras(1/2)the productof

ecomponentof thevoltage(b).

ntof/that isinphasewithE„ namely,

meproductis regardedasformedthrough

mponentofE,inphasewith I,thatis,by

of theimpedanceangle6itshould be

eequivalentto


/ h t t p : / / w w w . h





ksthealternativeformforPavexpressedin

ynow writethisresultin themoreexplicit

| cos(2ut-6)(53)

nasthereferencevector.

ortheinstantaneouspowersuppliedby

.4foranassumed6 =G0°.Weseethat the

ctionoftimeisthe sumofaconstantand adouble-fre-

gativeduringportionsofeveryperiodas is

roughcross-hatching.Theareaof each

resentsanamountofenergythatis being

thestorageelementsinthecircuit.

utthepowersuppliedto acircuitin the

hatit isnotrepresentedbya uniformflow

his flowhasapulsatingcharacterandthat,

bothdirections; thatistosay,it flowsfrom

urceaswellas fromthesourceintothe

nslossyelements(resistances),thenonthe

sintothenetworkthanis returnedtothe

ppliedpersecond bythesourcebeingPsvas

eextenttowhichthe circuitislossyis evi-

ghthevalueofcos 6,whichisunityfor 0=0

ordingtoEq.51 wenotethatcos6 becomes

ch,werecall,occurs atresonance(thatis,

herhand,forR=0, cos0becomeszero.

rveofFig.4 oscillatessymmetricallyabout


/ h t t p : / / w w w . h





averagepowersupplied),whilefor Z=R

ethezero axissothatthecross-hatchedpor-

sticallyindicatestheextenttowhichthe

mthesource,itis calledthepowerfactorof

forPavgivenbyEq. 50isreferredto as

ourcevoltageandcurrent.Exceptfor the

uctofavectorvoltageand avectorcurrent

uctofthe lengthsofthesevectorsandthe

enthem.Invectoranalysis(whichdealswith

vectorsbutislikewiseapplicabletothetwo-

mbersconsideredhere)thistypeofproductis

auseityieldsa scalarvalue(onehavingmagni-

powersuppliedtoacircuitfrom asource

begiven byone-halfthescalarproductof

rentatthe source.

relationtotheseries RLCcircuitofFig.2,

qs.45 and47through53,havingto dowith

utpowere, Xialone,obviouslyapplyat

earpassivenetwork,regardlessofitscom-

arepertinenttotheEqs.54, 55,and56of

reactivepowerandto vectorpower.

er;VectorPower

ntupontheresultsofthe previousarticle

esimplerones applyingtothecomputation

erethe powersuppliedissimplyequalto

currentat thesource.Apartfromthefac-

have abitmoreto saylateron,themost

hepowerrelationpertainingtoa-ccircuitsis

erfactorcos6.If thecircuitcontainsno

resistance,thentobesure thepowerfactor

emainingdifferencebetweend-canda-c

efactor1/2enteringin thea-ccase.In

eelementsarepresent,andtheymaketheir

orbingandreturningenergyduringeachcycle

yon theaverage.

ng asthesestorageelementsabsorbno

,theirpresenceor theireffectuponthecircuit

stto theonewhohasto payfortheenergy

gyactuallyconsumedmeansworkdoneby

yinthisargumentliesin theassumptionthat


/ h t t p : / / w w w . h




WER;VECTORPOWER

hastobe paidfor,orthatenergythat is

and thenreturned(undamaged)shouldcost

ompanieswhoarein thebusinessofsupply-

cejustifiablyfeelthattheyare entitledto

onlytemporarilyusedbythecustomerand

,becausethecompanyhasto gotothesame

nerateanddistributethe energy,whether

otwithstandingthefactthatonlythecon-

diminishesthecoalpile.

owtheenergythatisswappedback and

ndthecirc uithastobe kepttrackof.Al-

snotwell suitedasadesignationfor this

cepowermeansenergyflowandontheaver-

helessthetermreactive powerorwattlesspower

rencetothephenomenonwearediscussing.

curatelywhatismeantbyreactivepower

computedwiththesameexactnessasis

agepower),oneis ledrathernaturallytoa

roughreferencetothevectordiagram,part

agepowerisrepresentedasgivenby one-

hecomponentof/that isinphasewith E,.

choose(asanarbitrarydefinition)one-half

uadraturecomponentof/(thatcomponent

ldingthewattlessor thereactive"power"

rdingly,the componentofsourcecurrent

ource voltageissometimesreferredtoasthe

current.

veragepowersuppliedbythesourceis

Ej).Sincethereactivepower,as just

aginarypartof\(E,I),itseemslogical to

ty\(EJ)asa vectorpower.Itsrealpartis

suppliedbythesource (alsoreferredtoasthe

aginarypart(bydefinition)is thereactive

veoraveragepoweris denotedbythe

oneabove),andthe reactivepowerbyQav

heQofa resonantcircuit).Thuswehave

=\[E,I)(54)

ayalternativelywrite

|2X Y(55)

tual casetocomputethepowerper voltof


/ h t t p : / / w w w . h





epowerforanyother appliedvoltageis

multiplyingbythesquareofthat voltage.

powerthenbecomesparticularlysimple,

W)+jB(u)](56)

ensionally)theactivepowerPav(per

halftheconductivepart oftheadmittance,

avis one-halfthesusceptivepart.Theactive

rcuitclearlyisalwayspositive;thereactive

eeithersign,beingpositiveina capacitive

nductiveone.TheunitsofQavare called

olt-amperesreactive."

ternativelywrite

(57;

*)-jX(u)] (58)

owersuppliedtoanimpedanceZis appro-

entr atherthanthevoltageisknown.Thus

orpowerabsorbedbyanimpedanceperpeak

equalsone-halftheconjugateof thatim-

rrmsvaluesofvoltageandcurrent dis-

cle,the resultsexpressedbyEqs.56and 58

atthe factors1/2dropout.Thatis tosay,

dbyanadmittanceY perrmsvoltappliedto

Y;andthevectorpowerabsorbedbyan im-

passing throughitisnumericallyequalto

stingoftheinterconnectionofmany

tsaresimplysummedoverall thebranchesin

ectorpower.Thus,ifEkandIk arethecom-

entin branchfchavingtheimpedanceZkor

k orIk= EkYk,thenthetotalvector

h|%=\EI Ek \ *Yk (59)

endsoverallbranchesinthe network.

edefinitionofreactivepowergives one&

ue,andhenceprovidesa numericallysecure


/ h t t p : / / w w w . h




WER;VECTORPOWER

owercompaniescanbargain withregardto

rtheloanofenergyto supplystorageelements,

a verysatisfactoryframeofmindregarding

smysteriousreactive"power"thatisn't

tionwereturnto thecircuitofFig.2 and

e,(t) =E,e3atandi(<)=we getin

one-half,weget thevectorpower

2)

age|E |=|/ |/Cw;andbyEq.19

62 yields

JMV«v -Tav)(65)

stedin theresult

vepoweris proportionaltothedifference

rgystoredintheelectric fieldandthatstored

ughderivedhere forthesimpleseriesRLC

ds trueforalllinear passivenetworks,how-

dwithagivencircuitstore,on theaverage,

thentheymerelyswapac ertainamountof

weenthem,andthesource isnotcalledupon

onceithasreacheda steadyrepetitivepat-

isfact,see Art.6,Ch.10.


/ h t t p : / / w w w . h





state).Itis onlywhenFavT*vthat

s continuouslyplayedbackandforthbe-

rcuit. ThereactivepowerQ»visthus seen

enttowhichthe sourceparticipatesinthe

becauseitis proportionaltotheexcessin

icas comparedwiththemagneticstored

maybenumericallynegativeaswellasposi-

ssibleforone passivecircuittosupplythe

another.Such aprocess,whichrelievesthe

ntering intotheroleofan energylending

ustomerofthe burdenofpayinganadditional

icpowercirclesas"power-factorcorrection,"

ppropriatesincethereactivepoweriszero

ity,and viceversa.Reactivepoweristhus

a passivecircuitcansupply,andwe note

atelyinterpretingtheterm "power"inthis

riefmentionis seenthroughreferenceto

einstantaneoussuppliedpower.Weare

nizethat,whereastheactivepoweristhe

terminthe expressionfortheinstantaneous

tudeofthevectorpoweris theamplitudeofits

ingcomponent.

ffectiveValues

usoidalalternatingcurrentis accomplished

in whichthetorqueactuatingtheindicator

ionaltothesquareofthe current.Owingto

eelement,theactualdeflectionissteadyin

eof thetorqueandisproportionalto the

oussquaredcurrent.AccordingtoEq. 38,

rhasadeflectionproportionalto| 112.

chosenscale,thisinstrumentcanprovide

antity|/|2,or for|1|,or| /|multiplied

arrangedto read|1|2,thescaledivisions

hereas,if|/| or|1| timessomeconstantfac-

divisionswill needtobenonuniformlike

ngtoy= i2forequalintervalsin x.

eisprovided,andtheconstantfactorof

thenoneobservesfromEq.47 thattheloss

currentisgiventhroughthe productofR


/ h t t p : / / w w w . h




FFECTIVEVALUES353

erreading,justaswithd-c circuits.The

ntobe aquantitythatplaysthesamerole

tionofpowerl ossinthesinusoidalcase as

ntcurrentinthed-c case.Similarremarks

esince,by duality,allthatis saidabove

sintactuponinterchangeofvoltagewithcur-

dmittance(resp.RwithG andXwithB).

es| E|are,therefore,referredto as

soidalalternatingcurrentor voltage.They

entd-cvaluessincetheir power-losseffectsare

dwithequal valuesofconstantcurrentor

ssipatedina resistanceRbyasinusoidal

aluemaybe expressedasthesquarerootof

redcurrent. Thereforetheeffectivevalue

heroot-mean-squarevalue(abbreviatedrms

ntappliestovoltage.

wsthatthermsvalue ofasinusoidequals

result thatmayreadilybeverifiedinde-

s2x )dx

dcurrentare statedinconnectionwith

cuitwork,theyareusuallyunderstoodtobe

cetheseare thevaluesthatareread ona-c

torvoltagesandcurrents onecanhaveit

olsEand/ representeffectivecomplexvalues

byl/\/2).Since voltagesandcurrentsare


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e




chother,allinterrelationshipsarethe

presenteffectivecomplexvaluesoractual

ointhe discussionsofthepreviouschapter.

n thepresentchapter,theresultof having

ctivevaluesissimplythatthefactors 1/2

essionsforactive,reactive,andvector power.

itsuse effectivevaluesthroughout,and

giveninthem differfromthosegivenhereby

sttextbooksvectorvoltagesandcurrents

tivevaluesunlessotherwisestated,whilein

sof currentandvoltagevectorsarethe peak

yrepresent.If sinusoidsofonlyasingle fre-

most a-cpowerworkat60cycles persecond),

throughoutis possibleandconvenient.How-

sawkwardifnotimpossibletoadhereto in

uchasare metincommunicationsandcontrol

fdifferentfrequenciesmustsimultaneously

ttextis writtenfromthepointofviewthat

kforamorecompletestudyratherthanthe

strictedcase.

einTerms ofEnergyFunctions

theusualdefinitionsof impedanceand

Pav+j2u(V>v- Tav)(71)

admittanceas

avin Eq.72areassumedto beevaluated

mittanceY(u)is expressedexplicitlyin

nergyfunctions.A similarinterpretation

ceexpression73on thetacitassumptionthat

edforI =1ampere.Inconnectionwith the

sultsare onlyofnominalinterestsincethe

forY(u)andZ(w)inthis caseareevenmore

s72and 73,andsoit isonlythenoveltyof


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



NCEINTERMSOFENERGYFUNCTIONS355

pressedintermsofpower andenergythat

issignificantto mentionthatEqs.72and

elltolinear passivenetworksofarbitrary

onsforPav,Tav,andFavbeingcorrespond-

nemayseeagainthata conditionofreso-

hatis tosay,whentheaverageenergies

magneticfieldsareequal,theimpedanceor

ointreducesto arealquantity;thesystem

y,wheneverthedriving-pointimpedanceor

ginarypart,thenone mayconcludethatthe

eticstoredenergiesareequal;the power

ctivepoweriszero.

av,Vavare implicitfunctionsofthefre-

s72and73 arenotusefulin thestudyofY(u)

xceptinsomeveryspecialcircumstances.A

erationofthebehaviorofZ(w) inthevicinity

Inthesimple RLCcircuitconsideredhere,

2isaconstant.In moreelaboratecircuits

ratiosthroughoutthenetworkarealmost

cyrangeneararesonancepoint,and hence

ndsonlyuponthecurrentdistribution,is in

thevicinityof resonance.

=w0,Eq.63showsthat wecanwrite

frequencyin question,andthushaveinplace

nemayusetheapproximation

_ ,

ttreatedhere,thisexpression,aswell astheonegiven

ct,butin moregeneralsituationstheseareapproximate

ver,verynearlycorrectthroughoutanypronouncedreso-


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e




pedanceZ(u)thefollowingexplicitfunction

78)

hisexact)checkswithEq. 52as,ofcourse,

rX, throughuseofEq.64,checkswiththe

orthereactanceoftheRLCcircuitgivenby

esignificantfeatureaboutthisresultis that

allyforalllow-lossnetworks.

dillustratedtherein Fig.19,thehalf-

atedresonancecurveliewhereX= ±/?or,

cyincrementwbetweenthehalf-powerfre-

onancecurve)becomes

oundtobe expressibleas

1)

showninFig.1(applyingto thesimple

representativeofanylow-losssystemnear

TVcak,andEq.81canbewritten

le

thestored energymerelyswapsbackand

andmagneticfieldsandsothe peakvalueof

hetherexpressedelectricallyormagnetically.

qualsthe averagerateofloss(Pav)times

aracterizesthecriticalbehaviorofa low-

cemaybecomputedentirelyonanenergy


/ h t t p : / / w w w . h




MORECOMPLEXNETWORKS357

t83usefulbecauseit providesaninde-

omputationofthisimportantfigure ofmerit

o beusableinsituationswhereparameter

not feasible)butalsobecauseitprovidesan

ngwhatis meantbya"low-loss"or"high-Q"

whichthelosspercycle issmallcompared

totalstoredenergy.Inorderto obtaina

harpresonancecurve,one muststriveto

torage aspossiblerelativetotheassociated

gyFunctionsforMoreComplex

onsiderationhasseveralinductiveand

xpressionsforthetotalinstantaneousstored

nedthroughsimplysummingtherelations

ntbranches.* Symbolicallywemayindicate

"'EW] (84)

Sk I^ (85)

ectorcurrentinan inductivebranchhaving

summationextendsoverallinductive

Eq. 85,Ikdenotesthevectorcurrent in

elastance(reciprocalcapacitance)Sk,and

verallcapacitivebranchesinthe network.

nd85are TavandFavrespectivelyfor

tthesums yieldingthesequantitiesinvolve

esofthebranch currents,whilethesecond

hicharedouble-frequencysinusoids,involve

esofthebranch currents.Thesumsinthese

complexaddition(notmerelytheadditionof

he angleoftheresultantcomplexnumber

haseofthe pertinentsinusoid.Sincethe

alueshasaresultantmagnitudethati salways

tothesum oftheabsolutevaluesofthis set

ninductivebranchesishereassumedto beabsent.A

s restrictionisgiveninArt. 6ofCh.10.


/ h t t p : / / w w w . h





earthat theamplitudeofthesinusoids,

isingeneralless thanTxvorVavrespec-

onstantcomponentonlyifall squaredbranch

nditionthat existsinalllossless networks(for

on)andisnearly attainedinlow-lossnet-

a resonancefrequency.

timesmoreconvenienttodosointermsof

dofthebranchcurrents.Sincesucha branch

rrentbytheexpression

ewritten

"ty£CkEk2](8S)

eptforan interchangeofEwith/and C

predictedthroughuseoftheprincipleof

es

eresultsareappliedto aspecificcircuit,

IfwedenotebyEk thevoltagedropina

omputations89are pertinent.

tis Ik,andassumeE4=1 volt,thenthe

culationsforanassumedw=1radian per

ory

-1+jl(89)

E0=£i+E2=jl


/ h t t p : / / w w w . h




PLES"359

ilyhave

d

sin21(91)

have

|joule(92)

videntlynotalow-losssystem,for thestored

paredwiththe loss.Althoughthecircuitis

notequaltoFpeak=1. Thereisnopoint

ouldhavelittle meaninganyway.

mpedancefromtheenergyfunctions

/I\accordingtothevalues89.

hevalueofthe resistanceinFig.5to l/10th

9thenbecome

)

90wehave

1(95)

storedenergyfunctions,accordingto Eqs.

V= 25.25-25.25cos(2<- 11.5°)(96)

av =5watts(97)

etheresultsfora low-losssystemshould.


/ h t t p : / / w w w . h





so that

5(9S

0 XPav=2ir X5=10ir joules,andsothe

eitherEqs.81or83, yields

notextremelysharp,itis welldefined.

ordingtoEq.73,forthis casebecomes

e getZ=E0/Ii= 10,thussubstantiating

heserelationships.

eresistancetothe valueof1ohm,butadd

esasshowninFig.6. Thisprocedureshould

omputations101are pertinent.

yrelativetothel oss,andhenceyieldasharper

ntainresonanceatw= 1radianpersecond,

eedstobe 1/2henryasshown.Thisresult

firstcomputingthecurrentsinall ofthe

thepatternusedabove,andthennotingthe

eneededtomakeT*v =Fav.

onsappropriatetothiscircuit,assuming


/ h t t p : / / w w w . h




PLES

ve

thelosspercycle is3.14joules,itis clear

hsomewhatbetterthantheone inthefirst

a low-losscase.ThusTpeak=1is only

k=1.31.Ifwecomputea Qatall,it isbetter

ehave,usingEq. 73,

weget

titis astraightforwardmattertocompute

current distribution.Sincethelatterori ts

ntbedeterminedin thecourseofan imped-

outthat itisno moretedioustofindthe

energyfunctionsthan inthenormalmanner.

rgyfunctionscontainsmoreinformation.

tTav — Favissmallcomparedwith

canconcludethatthefrequencyconsidered

ance,especiallyif Pavissmallcompared

ughmakinga singlecomputationata


/ h t t p : / / w w w . h





areabletoperceivetheentirecharacterofthe

muchmorethanthevalueofZ atresonance

thenormalmanner,wemustcompute

nanceandplot acurve.Intermsofenergy

sameinformationfromasinglecalculation

quency.

gthesethoughtsisto callattentionto

ulatetheimpedanceoftheseries RLC

valuesofthe resistanceR,theinductive

tivereactance— 1/Cu,andthenetreactance

ivemagnitudesnotonlyenableus to

yinquestionisat ornearresonancebutthey

teroftheresonancecurve.Allthisinforma-

ofmakingacalculationatonly onefre-

ecircuitssuchasthoseshownin Figs.5and

toget thismuchperunitof computingeffort

ofthetechniqueof expressingimpedancein

forthisschemevirtuallyreducestheim-

ebasicform thatithasfor theseriesRLC

<,Pav=200 watts,Qav=-150vars(angleofZ is

erfactorofZ. (b)Findthecurrenti(t).

parallel withZwhich

actorequalto0.9lagging,

c)FindP„and QaT.

roftwo possiblevalues

eavoltageacrosstheloadof 100

gvectordiagramtoshow(i) loadvoltageinreference

sourceresistanceand lineimpedance,(iii)sourcevolt-

iveredbythevoltageE,whenC isnotpresentandwhen

art(a).Calculatethepercentageofthe powerlostinthe

othcases.


/ h t t p : / / w w w . h




t isexcitedbya sinusoidalvoltagesourcee,i—

tis foundtobe5 amperes(peak)andthepowerfactoris

itis excitedbyasinusoidalvoltagesourcee,j —

ctivepowerdissipatedis 200wattsandtheaveragereac-

L,andC.

stingofa 10-ohmresistanceanda20-ohminductive

espowerfroma 60-cycle-per-secondsinusoidalvoltage

stheload reads100volts.

ctivepower andreactivepowerabsorbedbytheload?

nnectedin parallelwiththeloadimpedancetoimprove

ctivepowershouldbedrawnby thecondensersothat

illbe unity?

pacitanceforthiscondition?

btainalaggingpowerfactor(current lagsvoltage)of0.9.

eofthe periodiccurrentwavesketchedinthefigure?

thereading ofadynamometerinstrument.)(b)Whatis

rrent?(This shouldcorrespondtothereadingofa D'Ar-

ecurrent isina 10-ohmresistor,atwhataveragerate

resistor?

usoidalalternatingvoltageEhavinganeffectivevalue

rto aninductancecoilwithconstantsas showninthe

sof thefollowing:(a)themaximuminstantaneouscur-

antaneousrateofenergydissipation(i.e.,energytrans-

ximuminstan-

rageinthe mag-

einstantaneous

nergystorageand

onare equal?

nthe ac-

me1voltacross

computethe

ranchesas well

a frequencyof

mputethequan-

thusestablishthat 10radianspersecondis very

ncy.ComputeQbyEq.81,and checktheinputimped-

oughuseofEq. 73.

entsan a-csystemconsistingofa60-cycle-per.second

mpedanceistakenaszerofor thisproblem,atransmission


/ h t t p : / / w w w . h





ohmsandL— 0.00172henry,andaloadimpedance

ntobe20 ohmsat60cyclesper second.Thepoweroutput

200watts,andthegeneratedvoltageisEt — 440volte

edattheloadis Pi0»a=6400watts.

tsequenceforcomputingthefollowingand obtaintheir

load= PFl,PFt,/(lml),^i0»d(rm«).(b)Whatarethe

softhe loadimpedance?

ardedas thesource.

,determineEiandIiforu — 1radianpersecond.

u,whatdoEiand hbecomeifall elementvaluesare

econd,whatareEi andhif onlytheinductancesare

(a),findtheaverageactiveandreactivepowerat the

ueofthe storedenergy.

der (d)becomeif|Et |ischangedto 10volts?

eaveragepowerenteringthe circuitattheterminals

werfactorisunity.Each impedanceabsorbsanequal

erfactorofthe impedanceZtatitsterminals2-2' is

mplitude7.

valuesoftheimpedancesZi andZj.

cedin parallelwithZtsoas tomakethepowerfactor

nation,whatwillbe theaveragepowerdeliveredbyE,

hat istheaveragepowerdeliveredatterminals2-2'?

opart(c)becomeif E,ischangedto 100voltspeak?

andEtareidealvoltagesources andZis alinear

nEi=»100/30°,Et=100/—30° (peakvalues),determine

energyflow(i.e.,from lefttorightor fromrighttoleft),


/ h t t p : / / w w w . h




dby eachvoltagesourceassuming(i)Z— 10ohms,(ii)

Oohms.

eaccompanyingsketchtakes50wattsat0.8 lagging

requencyis unity,whatare(a)the peakvalueofthe

ueofC, (c)thereactivepower?If theinputisa current

ue,whatisthe vectorpower?

is excitedatterminals1-1'(withterminals2-2' open)

esrms,theactive powertakenis50wattsat a0.555

nterminals1—1'arefreeand acurrenth=,20 amperes

nals2-2',thepoweris 300wattsat0.316laggingpower

cyisu ■10.

dsimultaneously,find(i)thetotalactive andreactive

theinstantaneousstoredenergiesTand V,(iii)the value

6 andcheckwiththereactivepowerfound inpart(i).


/ h t t p : / / w w w . h




ate

riumEquations

edanceconceptenablesonetodeal with

oblemsinamannerthatis identicalinform

e analysisofpurelyresistivecircuits(which

hercommonlyreferredtoasd-c circuit* *

sandcurrentsonlytheresistanceparameter

aw,intermsof complexcurrent,voltage,

othedeterminationofsteady-statea-ccircuit*

ermsofreal quantitiestotheformulationof

mayapplytosucha-ccircuitstudies thesame

zationthathas(inChapters1, 2,and3)

yto thesimplerd-cproblem.Morespecifi-

boutnetworkgeometry,aboutloopand

quilibriumequations,abouttheirformal

utionthroughuseofdeterminantsandother

ocitytheoremwhichonerecognizesbyinspec-

alformofsolution—allthesethings are

yingwithoutalterationinformto thea-c

t.6,Ch.2)that thesolutionofd-ccircuits

asetof equilibriumequationsoftheform

h— Ei

21I1— E2

=Ei

c"areabbreviationsfor"directcurrent"and"alter-

einga commondesignationforconstantcurrentorvolt-

usoidallyvaryingquantities.


/ h t t p : / / w w w . h




LBRIUMEQUATIONS367

arerespectivelythetotalresistancefound

ignatedasloop1, loop2,•• •,loopI;Ri2or

nthebranchescommontoloops1 and2or

umericallypositiveornegativeaccording

oopreferencearrowsareconfluentorcontra-

ch*);Ii ••.Iiare the"Maxwellloopcur-

closedcontours;andEi.•. Eiarenet

theseloops.

arsinthe sinusoidalsteady-stateanalysis

tanceparametersR,karereplacedbycom-

ers£,k,andtheE'sandI's arecomplexnum-

anceofaparticularf«* isthesameasthat

.Thusfa, f22,etc.arethetotalimped-

contoursofloops1,2,etc.,computedfrom

e,andcapacitanceelementsonthesecontours

ontourwerepresentedasaseparateproblem.

ethe impedancesofbranchescommonto

scriptsrefer, withthesameruleregarding

tothe R,k'sinthed-ccase.

arallelismbetweenthea-cand thed-c

nodebasis. Forthed-ccaseonehasthe

nEn=Ii

G2nEn=I2

nnEn=In

respectivelythetotalconductancecon-

whileGV2orG23,etc. areconduct-

ngnodes1 and2or2 and3,etc.(allcon-

tivesincereferencearrowsforthenodesrela-

econsistentin thattheyallpointfrom the

En arethepotentials(withrespectto

he respectivelynumberednodes;fand

esfeedingthesenodesand returningthrough

eon aplane,themeshes(asin afishnet)are chosen

owsareconsistentlyclockwise,thenallsignsarenegative.

roducevoltagevariablesthatarepotentialdifferences

tsetofnodepairsas isshownforpureresistancenetworks


/ h t t p : / / w w w . h




WORKSINTHESINUSOIDALSTEADYSTATE

thea-csteadystate,theonlyessential

replacedbyasetofcomplexadmittances

eremainingthesame.Thustin,17227etc.

ofthevariousbranchesthat convergetoward

dfromtheconductance,capacitance,and

esebranchesjustasonewouldthe resultant

combinationofthesebranches.Similarly

admittancesofbranchesjoiningnodes1and2or

eregardingalgebraicsignthat appliestothe

mthediscussionoftheanalogousd-cprob-

otalnumberofbranchesinthe networkand

es,thenthenumberofindependentnode

entKirchhoffcurrent-lawequations)is

dentloopsor meshes(numberofindependent

ations)is

ntvariablesontheloopbasis isI,andon the

eenthetwomethodsofanalysisin agiven

ichofthetwonumbersI ornis less,and

tcanbest berecognizedthroughexperience

toutthata branchmaybedefinedin

sicdefinitionofa branchistoconsiderit

L,or C),itispossibleto regardanyaggregate

parallelcombination)ofelementsasabranch.

illusuallyindicateachoiceinthis regard.

have thesinusoidalsteady-stateequi-

2

ntofthis setofequationsbyZand itsco-

dingto"Cramer'srule"(seeArt.2,Ch. 3)


/ h t t p : / / w w w . h




LIBRIUMEQUATIONS369

Et)(6)

eviationwemayi ntroducetheadmittances

essedby6 as

EkH+ VkiEi(8)

ntsthe currentthatwouldresultinthe

newereappliedinloop 1;thesecondterm

esult inthefcthloopif E2alonewereapplied

equantityyk\is thetransferadmittance

sthe transferadmittancefromloop2to

kk,whichisthe ratioofcurrentinloop fc

(assumingnoothervoltagesareapplied),

pointadmittanceofthe kthloop.Equation

of driving-pointandtransferadmittances

cttothe chosensetofloopsor meshes.

heEqs.5 issymmetricalaboutitsprincipal

ayinwhichtheelementsare defined,

),onefindsaccordingtodeterminanttheory

haty,k=ykl.Thus the.ratioofcurrentin

sthe sameastheratioof currentinloopk

tthatis knownasthereciprocitytheorem*

erballyis tosaythatther atioofresponseto

interchangeinthe pointsofexcitationand

asisonehas thesystemofnequilibrium

1

V2nEn=I2

VnnEn=/n

maybedenotedbyY anditscofactorsby

dsthesolutionin theform

YnkIn)(10)

Ch.3withrespectto resistancenetworksisdirectly

seoftheimpedanceoradmittanceconcept(specifically,

ofEqs.76,Ch. 3,bytheEq.17, definedhere).


/ h t t p : / / w w w . h





hH+zk Jn(11)

-pointandtransferimpedances,sincethe

amephysicalinterpretationasthoseinEq. 8,

ndvoltageinterchanged.

coefficientsof9expressedby17,* =if*,

sinceityields z,k=z*,.Thatis,the ratio

rcecurrententeringnodesis thesameas

estocurrentfed intonodek.Orthe ratio

invariantto aninterchangeofthepointsof

n.

connectionwiththereciprocitytheorem

mpedanceortoanadmittance—nottoa

io.Thusofthetwoquantities—excitation

beacurrentandthe otheravoltage.Onthe

a voltageandtheresponseacurrent;onthe

rue.Ontheloopbasisthe sourceisimped-

ntoanybranchof thenetworkhasnoeffect

fweconsiderthecurrentin loopsdueto

rnativelythinkofthevoltagesourceswitched

ethecurrentinloopk, itisclearthat the

ficientsremainunchanged.Ifweconsidered

ontobe acurrent,thenitsinsertioninto a

cuitthatloopbecauseoftheopen-circuit

urce.Switchingacurrentsourcefromone

gethephysicalcharacterofthe network,and

osethat thecurrentresponseinthealterna-

the sourcelocation.

,wheresourcesare currentsand,there-

shiftingofa sourcefromonenodepairto

kunchanged"impedancewise,"andtheratio

agebetweennodes)to excitationisun-

eofthetwonodepairs inquestion.Ifthe

econsideredtobe avoltage,orifthe response

t,onewouldagain findthataninterchange

onandexcitation(whichimpliestheremoval

ofthenetworkanditsinsertionsomewhere

ein certainofthecharacterizingadmittances

assurancethattheratioofresponseto excita-


/ h t t p : / / w w w . h




ICES371

,kofEq. 7orto thez,kofEq.12. Itmay

ovoltagesorto twocurrentsinsomevery

tgenerallyspeakingthereciprocitytheorem

anceor toatransferadmittance,notto a

nction.

s

5havethe detailedstructureindicatedin

e theso-calledresistance,inductance,and

heloopbasis.In writingtheseparameters

kit isexpedienttoassembletheirvaluesin

cterizedon loopbasis.Elementvaluesareinohms,

nd columnpositionsmaythenbeusedfor

on,withanobviousconservationofspace

amplein Fig.1willserveto illustrate.Here

R]=


/ h t t p : / / w w w . h





tionofamatrixelementidentifiestheparam-

[R]theelementinthe secondrowandthird

.Theprocessofwritingdownthese matrices

rkwithits chosenloops,referencearrows,

rlyindicated.

asisthe structureofthecoefficients

17)

rerespectivelythe conductance,capacitance,

parametersonthenodebasis.I naspecific

cterizedon nodebasis.Elementvaluesareinmhos,

rys.

downin matrixformbyinspectionofthe

2willillustrate.Herewefind:

=[G]

=[C]

x=[r] =|^J(20)

ya factorisequivalenttomultiplying

atfactor,andadditionofmatricesis carried

ntshavingcorrespondingrow-columnposi-

)


/ h t t p : / / w w w . h




nts(Eq.13)

22)

dcofactorsZlkthedriving-pointandtransfer

putedaccordingtoEq.7.

3)

nts(Eq.17)

24)

cofactorsY,kthedriving-pointandtrans-

mputedaccordingtoEq.12.

asons,theloopandnodebasesare regarded

msof theprecedingcompactformulation

ndadmittancesonemaystateinrathergen-

nderwhichtwonetworksAand Barere-

er.Namely,theyare soregardediftheset

identicalwith thesetofadmittancesy,kof

lcomeaboutif theloopimpedancematrix

calwiththenodeadmittancematrix[Y],

n turnrequiresthattheparametermatrices

kbeidenticalrespectivelywith[L],[R],[S]

onditionforthe lattersituationisobviously

ndentnodesn foronenetworkbeequalto

tloopsIof theother.

i llustratedthroughanexample.Sup-

wishestofindthat networkwhichisdualto

ncethisonehasthe node-parametermatrices

avetheloop-parametermatrices


/ h t t p : / / w w w . h





inFig.2.

s,henrys,and

edasbeingtheloop-parametermatricesof

3.

ointand transferimpedancesziitz2t.

2areequalrespectivelytothedriving-point

/h,

fFig.3.

ce

min

it-

b-

anch

the

nce

nce

min

22)foundforthenetworkofFig. 3through

heratioofcomplexcurrentin loop2tovolt-

whichisy12,is equaltotheratioof complex

ntfedintonode1 (thisiszi2)for thenet-

owtoDealwith It

f inductancesinrandomorientationwhose

sedsomehowtobelinkedsothat atime-

nductanceinducesvoltagenotonlyin that

he othersaswell.Theamountanddirec-

relativeto the

urrentisin

ycharacterized

ncecoefficient;

voltageappears

ntheone

ent,thecoeffi-

tualinductance.

osethatthe

byinductance

currentiiand

ent(instantof

/dtis+1.By

rentt'iinthe inductance1isincreasingin

onattherate of1amperepersecond(it

ally


/ h t t p : / / w w w . h




NDHOWTODEALWITHIT 375

edthroughtheapplicationof anappropriate

notinterestus atthemomentandis not

varyingcurrentinduces,first ofall,avolt-

selfwhichclearly_ mustbeavoltagedropin

t'isincethis inducedvoltageaccordingto

ethecurrentincreasethatis producingit.

tance1anincreasingcurrentin thereference

cedvoltagedrop whichisalsoin therefer-

denotethisvoltagedropby viwehave

n,whichis positive,iscalledtheself-induct-

nce)of coilnumber1.

dsetupby thecurrentiilinks theother

-varyingcurrentinducesvoltagesthere.

directionsthatarea prioriinnoway related

ceduponthe inductancesinwhichthey

voltageinducedincoil2 bythedii/dt= +1

makeeitherthetipend orthetailend of

ewithrespectto theoppositeend.Experi-

eterminewhichendis thepositiveone.If

duced voltageissuchasto beavoltage

I nthiscase,themutual-inductancecoeffi-1

2issaid tobepositivebcoan5e~thg"VoltageI

cindirectionas itwouldbeif itwereinduced

ell thatthissamemutual-inductance

derednumericallynegativeifwechangedthe

hich wecancertainlydoif wewishsince

enatwill.However,oncether eferencearrows

healgebraic signofthemutualinductance

xed.Itispositiveif apositivedi/dtinone

gedrop intheother(accordingto thearrows

edirectionsbothforcurrentsand voltage

ualisnegative.

ealgebraicsign ofamutualinductance

enin pairs.Thus,withcoils2, 3,and4open-

dt producedincoil1,we haveinaddition

givenby28


/ h t t p : / / w w w . h





s voltagedropsaccordingtothereference

coils.Themutual-inductancecoefficients

enumericallyequaltoand havethe

the voltagedropsv2,v3,t>4respectively.

that,althoughwemaybedealingwitha

terminationofanyone mutual-inductance

,involvesonlythetwocoils1 and3andis

ercoilsare presentornot(exceptasthe

mayphysicallyalterthe mediuminwhich

tedwithcoils1and3 resides).Thatisto

minethecoefficients£2i>'31,hi,etc.,experi-

tionisconcernedwithonepairofcoilsonly

presenceofthe others(excepttoseetoit

cuitedduringtheexperimentsothatthere

di/dt)'sexcepttheonespecificallyintended

asonthedeterminationofthemutual-induct-

egroupofcoilsi severybitassimple and

justtwocoils,becauseoneconsidersonly

othersaremeanwhileignored.

ncearrowsonthecoils,asshownin Fig.4,

-inductancecoefficientsiscompletelyfixed

tude.Specifically,ift'i,i2,i3,i4 arethecoil

re thevoltagedrops,bothwithregardto the

ws,thenwecanrelatethese currentsand

tions


/ h t t p : / / w w w . h




DHOWTODEALWITHIT

ybeintegratedwithrespecttotime,

+hiU

3*3+^24l4

3i3+hiU

+

rearefluxlinkages(sincetheir timederiva-

equationsmaybesolvedforthe coilcurrents

esbyanyalgebraicprocessapplyingtothe

nearequations(suchasthedeterminant

3^3+714^4

23^3+724^4

3^3+734^4

3^3+744^4

re denotedby

mericalcoefficientsfoundintheprocessof

.Forexample,ifthedeterminantofthe

dby

enbyCramer'srule

pletelyunderstandsthedetailsofsolving

atthemomentoflittle importance.The

s downhereisratherto beabletocallatten-


/ h t t p : / / w w w . h





n(throughwell-definedalgebraicmethods)

set ofmutuallycoupledcoils(Fig.4)in

s(specificallyinterms ofthevoltageinte-

yasonecanexpressthe voltagedropsinthese

ents(specificallyintermsofthecurrent

onein Eqs.30,theformerinEqs. 32.In

etheself- andmutualinductancesforthe

32thecoefficientsaretheself-and mutual

rthesamegroupofcoils.Thelattercoeffi-

ormerinamannerexpressedbyEqs.34 and

rminationofthe algebraicsignofamutualinductance.

fficientsininversesets ofsimultaneouslinear

reciprocalinductancecoefficientsy,kare

eciprocalsoftheinductancecoefficientsl,k,

tedina one-to-onerationalalgebraicmanner,

ssimpleandstraightforwardinitsapplication

mberofcoefficientsislarge).

ediscussionofhowthepresentrelations

settingupequilibriumequationswhenagroup

uchasthosein Fig.4isimbeddedin agiven

tionalremarksmaybein orderwithregard

ebraicsignsformutual inductancesinsitua-

ectionsofcoilwindingsand mutualmag-

chematically.Asituationofthissortis

preferredpathtakenbythemagneticfield

tangularcorestructure(whichmaybe the

andthe windingsofthecoilsaredrawn in

ognizesthedirectionsin whichtheyencircle

eleft-handwindingsoas tomakethe

e,currentinthiswindingincreasesinthe

ordingtotheright-handscrewrule,thefluxct>


/ h t t p : / / w w w . h




DHOWTODEALWITHIT379

directionshownbyitsarrow.Bytherule

hisaleft-handscrewrule becauseofLenz's

asingcoreflux ct>inducesavoltageinthe

makethe bottomterminalpluswithrespect

erencearrowontheright-handwindingas

einducedvoltagethereis avoltageriseora

thereferencearrowsshown,themutual

mericallynegative;itbecomespositive,

arrowoneitherwinding(notboth) isreversed.

plethattheplus-markedendsofthetwo

ngendsinthesensethat theywillalwaysbe-

usto-

duced

ging

lessof

n-

ends

wellas

tbe „„n .....,

othesigndetermina-

.Fortionofa setofthreemutualinductances,

ndap-

refertomarkcorrespondingwindingends

plus signs,andthisis awidelyaccepted

hemeofrelativepolaritymarkingcannot

odificationwhenmorethantwowindingsare

magneticstructure,asthefollowingdiscussion

llshow.Ifweassumethetop terminalin

ithrespecttothebottomone,currententers

hearrowdirection,thusproducingafluxthat

reofwinding1 anddownwardinthecores

theirwindingdirectionsrelativetotheir

s thebottomendsofcoils2 and3thatbecome

placeadotat thetopofcoil1, andcorre-

msof coils2and3.If wenowmovethe

andmakethebottomterminal(the dot-

seethatfluxincreasesdownwardinthe core

dinthecoresofcoils1 and3.Thusthetop

oils becomepositive.Forcoil1thisterminal

forcoil 3itisn't. Therefore,itbecomes

minalscaningeneralindicaterelativepolarities

cpairofcoils. Onewouldhavetouseadif-


/ h t t p : / / w w w . h





airofcoils 2and3from thosethatarealready

pairingthem separatelywithcoil1.

ngrelativepolaritiesof mutuallycoupled

husseentobecomeprohibitivelyconfusing

areinvolved,thedeterminationofaset of

cecoefficientsconsistentwithassumedrefer-

leandunambiguous,asalreadyexplained.In

learlyfindallthreemutual-inductancecoeffi-

llynegative.Oncetheseareknown,thevolt-

oupof coilsisunambiguouslywrittendown

2.

mplecaseofjust twomutuallycoupled

dinductancecoefficientsbedenotedby£n,

pererelationsread

)

t)

onsrespectivelybyii andi2andadd,we

H— +hiiz— +£22*2— (37)

ywriteas

1*2*1~T"£22*22

)(39)

yunderstandablephysicalfact,namely,

owerabsorbedbythepairof coils(viii+t>2i2)

changeofthe energyTstoredinthe asso-

elatterbeinggivenbyexpression39.Alge-

homogeneousandquadraticinthecurrent

s aquadraticform).Physicallyitisclear

matterwhatvalues(positiveornegative)

have.Mathematicianshavefoundthatthis

conditionsonthecoefficientsl,k.Specifically

tobe apositivedefinitequadraticform,it


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



S

thatln >0,l22> 0,andinaddition

coefficientforthepairof coilsinquestion,the

ciatedstoredenergybepositivefor allvalues

othecondition

essedby\k\ =1,whichis approachable

pairofphysicalcoils,is spokenofasacondi-

closecoupling.Physicallyitrepresentsa

uxlinksallofthe windingsofbothcoils.If

Eq.42) isderivedfromthestandpointof

sarrivedat onthebasisthatthe stateof

stlyanupperlimit. Adifficultywiththis

on43is thatitdoesnot lenditselftogen-

odbaseduponstoredenergyis readilyex-

upledcoils.

reasoningleadingfromEq.36toEq. 39

gyis ingeneralexpressibleas*

+hniiin

1+hnhin

In,h2,etc.are positiveinanycase,thecon-

areexpressedby statingthatthedetermi-


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e




ughcancelationofthefirst rowandcolumn,

umns,thefirstthreerowsandcolumns,etc.

s)arepositive.Althoughitis notthepur-

sionto godeeplyintomattersofthis sort,it

ointout (whereverthiscaneasilybedone!

eforextendingourconsiderationstomore

EquationsWhenMutual

lypresentedin termsofaspecificexample.

henetworkofFig. 7,forwhichtheecjui-

e equilibriumequationsaretobefoundon theloop

valuesareinohmsanddarafs.The coupledcoilsare

andmutual-inductancevaluesinmatrix46.

dontheloopbasis.Sofar astheresistance

matricesareconcerned,thereisnonewprob-

eweneedconcernourselvesonlywiththe

ceparametermatrix.

nthethreemutuallycoupledcoilsLuL2, L3,

rowsindicated,shallbecharacterizedbythe

cematrix.

uctanceofhiis2henrys,themutual between

soforth.If thevoltagedropsinthesecoils

hen, sincethecorrespondingcurrentsare

ndi2,wehave


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



UMEQUATIONS

— +3—

= 25— (47)

— +7—

edroparoundloop1 isvi+ v2,andthat

3.FromEq.47 thisgives

48)

cematrixisseentobe

equaltoL21servesas apartialcheckonthe

pleexampleonthenodebasis.Letthe

g.8. Hereonlythemethodoffindingthe

e equilibriumequationsaretobefoundon thenode

valuesareinmhosandfarads.The coupledcoilsare

andmutual-inductancevaluesinmatrix50.

trixforthenodebasisneedbediscussedsince

esentsnonewfeatures.Thethreecoils

edto bemutuallycoupled,thematrixgiving


/ h t t p : / / w w w . h





ualinductancesbeing

sof mutualinductancesarefixedrelative

thecoils.

dthematrixwiththe reciprocalinduct-

venbyEq.35.Denotingthedeterminantof

orsbyA,*,wefindA =10,and

aredenotedbyin, ii2,*/3,then,sincethe

esarerespectivelythetimeintegralsofei,

accordingto Eqs.32

e2)dl+0.2je2dt

e2)dt-0. 2Je2dl

e2)dt+0.2je2dt

tdivergingfromnode1is in+t'jj, and


/ h t t p : / / w w w . h




ANCESFORLADDERNETWORKS385

s —1;2+113.From53this gives

0je2dt

8 J"e2dt

uctancematrixonthenodebasisisseen tobe

ualr2iservesasa partialcheck.

PointandTransferImpedancesfor

onwithpureresistancenetworksin Art.3

ofnetworkthatoccursfrequentlyinprac-

ncedladdernetworkshowninFig.9. The

areanytwo-terminallumpednetworks,the

ernetworktowhichtherelations56-72are relevant.

nentadmittancesandthez's theirimped-

withz'sarereferred toastheseries branches

eledwith y'sastheshunt branches.

s interestedinthedistributionofcurrent

evariousbranchesofthisnetworkin the

thoughthegeneralmethodofanalysis

articleis,of course,applicableinastraight-

sthatamoredirect procedure(followingthe

laddernetworks)canherebe usedwith

computationaleffort,asalreadyillustratedby

hepreviouschapterandfurtheremphasized

s.


/ h t t p : / / w w w . h





•.,Ei,E2,• ••,asindicatedin thefigure,

sofvoltageandcurrentas usuallydefined.

ceofrelationsisevident:

to57,then theresultingrelationtogether

nuinginanobviousmanner,onesuccessively

rrentsexpressedinterms ofthesinglequan-

hus ultimatelyexpressedintermsof

termsofEi orli—onehasallvoltagesand

msofEi orIi.It islikewiseclearthatthrough

onemayobtainallvoltagesandcurrents

one voltageoranyonecurrentthatwe may

vesubstitutiontakesthefollowingform:

Es(66)

z62/9+zs2/9+l)E5 =AE5(67)

zm+ y5zSyg+2/7zs2/9

s(68)

(70)

3z4+z2 +z4)B]E5(71)

aM+(2/1z22/3z4+^1z2+ 2/1z4+2/3z4+1)B]ES


/ h t t p : / / w w w . h




ANCESFORLADDERNETWORKS387

nytransferratio,suchasEs/Ei orEs/Ii,as

ceEi/Ii,mayreadilybecomputedinterms

s.Incidentally,it isworthnotingthat,if

dthe y'scapacitances,thenallofthebracketed

plepolynomialsinthecomplexfrequencyvari-

cedladderhavingthetransferrelation givenbyEq.77.

nsiderthenetworkofFig.10.Herewehave

s]E3(75)

+L2C5+L4C5)s2+1]E3 (76)

C3+C5)C,+(L4C,+L^Cds3

7)

regivennumerically,theseexpressions

t,andtheprocedurelosesthestillsomewhat

at,intheaboverelationships,mayleavethe

rableimpressionregardingitsbrevity.He

ever,thatthecomputationalmethodsug-

directthan theformaloneinterms ofmesh

nts.

anceEi/Ii,analternativemethodof

elpfulinsomepracticalsituations.This

pleruleregardingthecomputationofimped-

onnectedinseriesorin parallel.Withref-

sat theright-handendoftheladder,

tsalternatelyinseriesandinparallelas indi-

chisconstructedbystarting atthelower

rdtheupperleft-handendin areadilyunder-


/ h t t p : / / w w w . h





lythisformforthedriving-pointimpedance

tiondevelopment.Itisespeciallyusefulas

ninproblemswherethenetworkisto befound

venandthe impedancetobefound)since

edancesandadmittancesareplacedinevi-

pmentisobtained.Inanalysisproblemsit

ghtforwardcomputationalprocedure.

mmetryinStructureandSource

Circuits

on,transmission,anddistributionsystems

calsituationsoneencounterscircuitsthat

trywithregardtoboththeirgeometrical

uesaswellas theirmodeofexcitation.Such

eferredtoaspolyphasesystems,theterm

otas theusualdesignationoftimephase

tooneofthestructurallyidenticalpartsof

symmetricalwholemaybedecomposed.

work,accordingtotheseideas,isillustrated

e"ofitisshownin Fig.12.Thethreesinusoi-

einternalimpedancesaredenotedbyZ„are

rmtime-phasedisplacementbetweenthem;

20a,E3 =£2/zbl20° (79)

hase displacementsbeingeitherconsistently

gative(ifnegative,theconventionistosay

sitivesequence;if positive,anegativeone).


/ h t t p : / / w w w . h




ymmetricallyorientedvoltages(Fig.14)or

sistingofn identicalandsymmetrically

ceshaveequalmagnitudesandphasedisplace-

sor 360/ndegrees.

tryinthecircuitof Fig.11itfollowsthat

wiseareidenticalexceptforauniformphase

onnectedthree-phasecircuit.

,incidentally,thepointsnandn' areatthe

seenfromthe factthataconnectionofnand

hoff'scurrentlawthecurrentinit equals

s zero.Inpracticethe"neutral"points

ected,sincetheeffectofanaccidentalunbal-

mentthen

gunbalance

cethe

helink

rent,its

alanced

aterial.

sonecan

ughcon-

thesystem

olving

oritscur-

wntheexpressionsforJ2 and/3asIi dis-

nd240° respectively(ineithera laggingor

ngtothatapplyingtothesourcevoltages).

sentation


/ h t t p : / / w w w . h





earrangement,insteadofhavingthe

1,is inthedeltaconfigurationshownin

htherest ofthecircuitofFig.11, thedelta

sitselfless

per-phase"

necan,

hewyear-

entdelta

ofTheVenin's

rcuitvoltage,

nd2,must be

ds to

c )(80)

encesystem,

yields

-

dthree-

ceinthe

o-

are

egeometryin

agesineithersequenceare relatedinan

ofthetwosourcearrangementsis con-

,ifthesourcesaredead,the impedancebe-

Fig.11

mdiscon-

13itis

enceone

ofthe

nverted

alent

may

.*

hen

ced)

analysis

strating

qs.80 and

equivalence

t.

-wyetransformationrelationsdiscussedinArt.4 of

rksareformallyapplicabletoimpedancesandadmittances.


/ h t t p : / / w w w . h




ughthediscussion ofmoredetailedaspects

llsoutsidethe presentstudies,itisappro-

gnificantfeaturesofpolyphasesystems

d energyrelationships.

mple,interestingtonotethat itismore

cpowerovera three-phasetransmission

seline.L etussupposethata totalamount

ttedata voltageV(conductortoconductor

c onductortoneutralinthe polyphaseline).

factorcondition,thecurrentinthe single-

etotal linelossesare2I2R= 2P2R/V2

er lineconductor.Inathree-phaselinethe

=P/3Vbecausethetotal powertransmitted

erphase,or,fora giventotalpower,each

.Forthesameresistanceperconductoras

etotal linelossesare3I2R= P2R/3V2,an

thasl argeasthatforthe single-phaseline.

e fairsincethethree-phaselinerequires

dwithonlytwoconductorsforthe single-

ee-phaselinerequiresoneandone-halftimes

struction.Afaircomparison,therefore,

lamountof copperforbothlines.Onthis

nductorof thethree-phaselineisthree-halves

e,andthe totallossesofthethree-phaseline

esthevaluecomputedabove.Thusthethree-

ne-fourthinsteadofone-sixthaslarge as

ine,whichis stillarespectableimprovement.

theonlyfactorthat suggeststhepractical

m,sincethereare otherschemeswherebythe

issioncanbeimproved.Oneofthe further

sesystembecomesevidenfyfromaconsidera-

eouspowerexistinginsuchacircuit.Thus

he instantaneoussourcevoltageandcurrent

11| cos(ut+ct> )

deliveredbythissourceis

os(ut+ct> )

>)]


/ h t t p : / / w w w . h





evoltagesandcurrentsaregivenby the

orretarded)by120° and240° respectively;

ntity(ul)in 83isreplacedby (ut±120°)

ely.Thecorrespondingexpressionsforin-

ephasesare,therefore,thesameas piin

replacedby(ut±120°) and(ut±240°).

uspowerbecomes

e3i3

ut+ ct>)+cos(2w<+ ct>±240°)

inthisexpressioncancel,wehavesimply

6)

hisresult isthatthepulsatingcomponents

alizeeachother, sothattheneti nstantaneous

teadycomponentalone.It issimplythree

owerperphase.

chinerythisfeatureresultsin asteady

taininga pulsatingcomponent.Theprac-

dissignificant.

racterizedbythe matrix

ductancewhentheyareconnectedin thewaysshownin

d).

—rWtP>«—0


/ h t t p : / / w w w . h




ductancematrix,findthenetinductanceofthe indi-

mbinations.

rrentin theseriesconnectionisoneampere,compute

parateinductances(inweber-turns)andthetotalflux

lenergy(injoules)storedin theassociatedmagneticfield.

xlinkagein theparallelconnectionis1 weber-turn,

e separateinductancesandthetotalcurrent.Compute

gementofinductancewindingson acommoncoreas

dtheir variouspossibleinterconnectionsasshownin

chself-inductance-2; eachmutualinductanceinabso-

enet inductanceineachcase.


/ h t t p : / / w w w . h





cifiedintheprecedingproblem,findthe reciprocalin-

rixforthenode-to-datumsetofnodepairsindicatedin

atrixconstructanequivalentcircuithavingonlyself-

forthegraphshownin sketch(b).

uctanceandelastancematricesontheindicatedmesh

uctancesareinvolved.Findtheirvaluessuchthat the

metermatricesfoundabove.


/ h t t p : / / w w w . h




dreciprocalinductancematricesonanode-to-datum

uesofthe capacitancesshowninthe sketchare1farad

cematrixisthatgivenin Prob.5.

citancesandonlyself-inductancesthathasthesame

umbasisastheone inProb.6.Assignall elementvalues

elementvaluesare inhenrys,farads,andohms.Show

ork,regardlessofthenumberofmeshesinvolved,the

yshastheform

reactiveelements(capacitancesandself-inductances)

etworkgiven.

estobe unity.Showineachcasethatthe transfer

a heuristicbasisthroughuseofthe resultsofProb.8and

chisasymptoticallyvalidfors—> 0andanotherfor


/ h t t p : / / w w w . h





ckthroughevaluationofeachtransferimpedance.Show

tscanbeusedtoevaluatethe constantmultiplier.

s,henrys,andfaradsare allunity,andnomutual

UsingtheprincipleslearnedinProbs.8and 9,predict

ericalevaluation.

esare1-ohmresistances.Forthechoiceofmeshes

oopresistanceparametermatrix[R].Nowrevisethe


/ h t t p : / / w w w . h




meshcombinations:(1+2),(2+3), (3+4),(4 +5),

ix[R].Arethecorrespondingequilibriumequations

hemeshcombinations:1,(1+ 2),(1+2 +3),(1+2

+5),find[R]and statewhethertheassociatedequations

ketchbeloware inohms,henrys,anddarafs.For

shes,writedowntheloop-parametermatrices[R],[L],[S].

stsasecondnetworkwhosenode-parametermatriceson

S]

k,andindicateallofits elementvalues.Howmany

dtocharacterizethegivennetworkona nodebasisorits


/ h t t p : / / w w w . h





sketchhavevaluesas specifiedinProb.2.Thecapaci-

Forthe indicatedchoiceofmeshes,findtheappropriate

writetheequilibriumequations.

1faradeach.Theinductancesareasspecifiedin Prob.

uationsonthe indicatednode-to-datumbasis.

esare1faradeach.Theinductancevaluesarespecified

rvoltagevariableschooseei=»i,ei =v^,es=v%,and

ilibriumequations.

esareinfarads.Theinductancesareasspecifiedin

tialequationwhichdeterminesintermsof ei.


/ h t t p : / / w w w . h




alanced,positive-sequence,three-phasesystem.The

dareinohms.If |Ei|= 100volts,whatmust|Ea |be?

ofEarelativetoE\;of EtrelativetoE2;of Ecrelative

urrentsI\,It,I3 inmagnitudeandphaserelativetothe

currentsinthe delta?

rrentsourcesandresistancesi nsketch(b)is tobe

theterminals1, 2,3)tothe balancedthree-phasedelta


/ h t t p : / / w w w . h





ancesshowninsketch(a).Findthevalues ofRandIi, It,

hroughconvertingthecurrentsourcesinthewyeinto

esinserieswiththeirrespectiveresistances,obtainvoltage

entequivalenttothegiven deltaarrangementandcheck

ilowattsofpoweraretransmittedsinglephase ataline

sandunity powerfactorforadistanceof25 mileswitha

watts.Ifthesameamountofpowerweretransmittedthree

o-linevoltage,thesametotaltransmissionline copper,the

powerfactor,whatwouldbetheline losses?Whatwould

lagging? Whatwouldtheybeifthe totaltransmitted

s,allotherquantitiesremainingthesameas inthefirst

semotortakes12kilowattsat0.8lagging powerfactor

ne.Computethelinecurrent.If thismotorisfed from

ree-phasefeederline,andeachline conductorhasaresist-

thelinevoltageat thetransformer,andthevoltagedrop

Computethetotalfeederloss.


/ h t t p : / / w w w . h




with

ntBehavior

AlternatingExcitation

ing withstepfunctionandimpulsere-

eldsthebehaviorfollowingthesudden

opriateswitchingoperation)ofavoltageor

tantvalue.Thecircuitbehaviorundersuch

mesreferredtoasa"d-c"transient,theletters

directcur-

theconstant

nafteritsin-

wthatwe

othesteady-

mecircuitsfora

ogicaltocon-

lowingtheFia VRelativetothedeter-

ana-cswitching

oid:that isto.. .

tthis pointto

tionofso-called"a-c"transients(theletters

nforthewords"alternatingcurrent").

othingessentiallydifferentaboutthedeter-

ientsascontrastedwiththe d-ctransients

memethodsofanalysis apply,butsome

sufficientlydifferentandinterestingtojustify

nthediscussionofthis topic.Animportant

erstandingoftheimpedanceconceptandof

uency.

mpleRL circuitshowninFig.1,but this

atione,(t)assumedtobe asinusoid.The

t =0,thusinsertingthe sourceandper-

ist.Itisthis currentfort> 0thatwewish


/ h t t p : / / w w w . h




STEADY-STATEANDTRANSIENT

ort> 0reads

requencyofthevoltageexcitation.Apar-

tothediscussioninArt.2 ofCh.5,isgiven

hemoment(asjustifiedbythe discussion

bstitutiongives,aftercancelationofthe

(3)

tof thecompletesolution.Forthetransient

nction,weconsidertheforce-freeorhomoge-

n(asdiscussedinArt.2 ofCh.5),namely:

otdependuponthe excitation,itisthesame

ad-ctransient,and,hence,thecomplementary

fthe solutionisthesamein formasdis-

pfunctionorimpulse response,thatis,

sticvalue(complexnaturalfrequency)given

btainedthroughadditionofthe steady-

ransientpart,Eq.7,thus:

ng step-functionresponse,theintegration

romtheconditiontobe satisfiedattheinitial

texample,thenet currentmustbezeroat


/ h t t p : / / w w w . h




THALTERNATINGEXCITATION403

ethatthetransientamplitudeA asgiven

theinstantaneouscurrentdemandedbythe

lueis thediscrepancybetweenthesteady-

uit'sresourcesatthis moment,sincethe

ycurrent.Aspointedout inArt.2of

estherole ofashockabsorberin thatit

nfromtheinitial stateofthecircuitto the

ngfunction.Itshouldalso benotedinthis

ormofthis transition(i.e.thetransient

bythecircuit(notbythe natureofthedriving

same,regardlessofthe formofthedriving

c).Onlytheamplitudeofthetransient

edrivingfunction,andmerelyuponthede-

alinstant.Inotherwords,the onlydiffer-

ientandad-c transientisinthe expression

onalformis thesame,andthemethodfor

me.

,asgivenby Eq.10,ishoweversome-

ausethereisinvolvedanadditionalparam-

ed-ccase,namely,thetimephase ofthe

is,theangleofthe complexvoltageampli-

tevalueforthisamplitude,onemayencoun-

accordingtotheassumedphaseoftheexcita-

pectedaboutthisresult,since thesteady-

ngsinusoidal,passesthroughavarietyof

hperiod, andthetransientamplitudeAis

valuethissinusoidalcurrentfunction(par-

initialinstant,whichisanymomentwewish

antthesteady-statecurrentfunction

henthetransientamplitudehasthelargest

her hand,theinitialinstantis chosentofall

ady-statecurrentfunctionpassesthrough

plitudeiszero, andthereisno transient.

0,onecanreadilyinvestigatethesecondi-

thevalueofthe transientamplitudeis

ofthecomplexquantityenclosedinthe


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



TEADY-STATEANDTRANSIENT

videntlythis realpartiszero iftheangle

±ir/2 radians,anditisa maximumwhen

leequalstheangleof E,minustheangleof

sthephaseof thevoltageexcitation,which

the choicemadefort= 0,theinstantof

eptfor thefactorL,thequantity(sp — Si)

ourcircuit.Thesketchin Fig.2aidsvisual-

ngle6 initsdependenceupontheparameter

exapplied

intonthe3

asinusoid

oltage(angle

erthanthe

ueofA inEq.

rminEq.10

stancesthere

onthatthe

ion(thefirst

= 0sothat

snodemand

alinstant.

isequalto

enthesteady-statecurrentfunctionhas its

att=0, andtheassociatedtransientterm

mum.

turnsoutthatLsp =juL(theinductive

edwiththeresistance Rinthecircuit, a

telybeexpressedbywriting:1 Si|<SC||.

stheimpedanceangle6,accordingto Fig.2,

llynotransient thenoccursifthe angleof

ageexcitationis

nent

c uitin

tage

sw t

sientresultsif theangleofEis ±ir/2;that

-=f\E|sin ut

areexactlyoppositetowhatone might

sertionofthe voltagefunction11att= 0

tialvoltagejump(fromzeroto |E|),while


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e




n12 att— 0introducesnoinitialvoltage

ero fort=0). Sincethevoltagefunction11

an initialshock,whilethefunction12does

toconcludethattheexcitation11willcause

the excitation12willcausenone.The

ctlythe reverse,however,sinceitisthe

enthesteady-statecurrentdemandmadeby

chthecircuitcanprovidethat determines

t.Underthe presentassumptionthat

ntinthe circuitofFig.1followingswitchclosure.

currentlagstheappliedvoltagebyessentially

ctionwhene,(t)isacosine,andviceversa.

rrentisasine function,sothatitis zerofor

rrentdiscrepancyandhencenotransient.

te,(t)bea cosinefunction,notasine.

sein ahighlyreactiveRLcircuitunder

mumtransient.Showndottedarethesud-

whichisa sinefunction,thesteady-state

ng90° behinde,(t)andhencehavingits

att=0,and thetransientcomponentof

pleexponentiallydecayingfunctionwith

magnitudebut oppositeinsigntothe

esum ofip(t)andt'n(<)is thenetcurrent

e.Itstarts fromthevaluezero,risesin its

ostdoubleits normalamplitude,andin

duallycomesclosertothesteady-statefunc-

(t).


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e




urrent i(t)asthoughit werethefunction

ottedcurve t'o(0asanaxis.Since toW

hetime axis,i(t)smoothlymergeswith

ydamping(R=0),io(t) maintainsthe

becomesthecosinefunctionip(t)elevated

meaxis;the netcurrentthenoscillatescon-

ddoubleitsnormalsteadyamplitude.

nnotoccur inapassivephysicalsystem,

htlydampedcircuitduringa shortinterval

ncloselyresemblethislimiting behavior.

oexploitthesolutiongivenby Eq.10for

valuesofsp andthusgaina betterunder-

omplexfrequencyas wellasofthe imped-

on tothenaturalfrequency(s= Si)ofthe

g.2, whatweproposetoconsideris to

axis andwanderintothelefthalf-plane.As

tationfunction

soid,andbecomesinsteadadampedsinusoid.

=|E |e^,Eq.13yields

4)

valueofa (apointspin thelefthalf-plane)

nentiallydecayingsinusoidwithinitialampli-

izeis thatthecurrentresponseas given

sumedexcitationvoltageoftheform13 with

s wellasitdoes foranspvalueon thejaxis.

llthebehavioroftheseries RLcircuitfora

e,(t)equaltothesinusoidalfunction with

a?nplitudeasgivenbyEq.14.Wecan,inci-

umericallypositiveas wellasnegative,

positiveavalue representsaphysicallyun-

steady-state"portionoftheresponsethen

momentistheresponse10for anspvalue

thenwe canconsiderwhathappenswhen

ponSt. Physicallythisconditionisoneof

thecomplexfrequencyofthedrivingforce

hthenaturalfrequencyof thecircuit.In

nceconditioninthe simpleRLCcircuitas


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e




ysp= jwofthesinusoidalsourcehassub-

sthe imaginarypartofthecomplexnatural

heimpedanceforthisconditionbecomesa

tivelysmallamountofdampingsothatthe

ieslienearthejaxis) andthecurrentresponse

eFigs.16and18of Ch.6).

mplexfrequencyofthesourcetothej axis,

ecoincidentwithacomplexnaturalfrequency

aycomecloseto it.Thatisto say,solong

oidaldriving forcewithconstantamplitude,

selytoaconditionof perfectresonance,but

weconsidera dampedsinusoidaldriving

ssibletoachievecoincidencebetweenthe

dthatofthe circuitandthusevaluatea

are,ofcourse,dealingwithadegenerate

ralfrequencySihasno imaginarypart,and

encytooscillate.Neverthelessoneshould

spwith$i asaresonancecondition,anditis

valuateEq.10forthis situationasastepping

ofananalogousconditioninvolvingthe

anceZ(sp)=L(sp— Si)becomeszero,and

infiniteamplitudes.Therelationforthe

determinateforma>— w.Thismeaningless

e donotrushheadlong,soto speak,intothe

etsp— Si=8and regard8asa small

matelyallowtobecomezero.Wethus find

ncewewishtoevaluatethequantitywithin

allvaluesof8, itisappropriatetowrite its

s:

w obvious,andEq.15isseen toyield


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e




ritten

areshownrespectivelytheexcitation

thecurrentresponsecorrespondingtoEq.18.

hecircuitofFig.1 whentheappliedvoltageisthe

wnin(a).

ns

n

eries

e

S

hefactthat the

ervesthatthe

spiteofthe

ro,sincethe

nabletosus-

let alonean

o thean-

gtheRLCcir-

oltageexci-

edto havethe

wemakeno

especific

ed,whetherpure

essingequilibrium

eResign onthevoltageexcitation,sincewe

standsandtaketherealpart ofthesolution


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e




arintegralwecanwrite

q.19andcancelationofthefactore*** yields

)

ninArt.5 ofCh.6,wecanthus write

— ad

2=1/LC(23)

tion(transientpartofthesolution)isan

ousequationcorrespondingto19,namely,

eneralformas forstepfunctionorimpulse

atis:

itutionintoEq.24gives

thecomplexnaturalfrequenciess=siand

ns23.Hencethecompletetransientpart of

t+ A2e'* (28)

statecurrentamplitudeJgivenbyEq.22.

he capacitancecharge

" '(29)


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e




0)= g(0)=0],Eqs.28 and29yield

thesegive

Si)

dtaking therealpart,wehave

ecialformsofthis resultmaybeconsidered

citationwithsteadyamplitude(sp= jw).

cordingtowhetherthegivencircuitis assumed

ghlyoscillatory,whetherthefrequencywof

ecomparedwiththenaturalfrequencywi,

toit.Further,thesevariousconditionsmay

haracteristicallydifferentphaseanglesofthe

witchclosure.Allof theseresults,manyof

eirappearanceandsignificance,arecon-

venbyEq.32.

etoenterupon anelaborationofsuch

urselveswiththeconsiderationofthecondi-

orwhichsp= Si.Beingmindfuloftheanal-

heRLcircuit,weletsp =Si+8 wherever

=Sileadsto difficulty,andtentativelycon-

ity.Assumingalsothatthecircuitis highly

ndoj<i~w0,we have

33)

d


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



ITHALTERNATINGEXCITATION411

16 andconsiderationofthelimitS —* 0

sumingazero phaseanglesothatE isreal,

+— )(36)

,thesecond terminthisexpressionis negligi-

so farasthegeneralcharacterof theresult

ecircuit ofFig.5whentheapplied voltageisthedamped

that theenvelopefunctionsarethesameas thetotal

withtheotherapproximationsmadesug-

thefirstterm maybewritten

37)

ormasthat fortheRLcircuit (Eq.18)

o0lwhichgivesthepresentresponseits oscilla-

voltageexcitationand ofthecurrent

ctivelyinparts(a) and(b)ofFig.6. Again

thatalthoughtheimpedanceZ(sp)becomes

eady-state"amplitude/(Eq.22) isinfinite,

meaningful,asisalsothe conceptofcomplex

ampleplaysits rolewithoutreservationsof


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e




houttheconceptofcomplexfrequencythe

blemwouldnot havetheabovestraight-

uldadiscussionofthephenomenonofreso-

entofthisconditioninits fullestsenseboth

ly.Itiswiththefurther elaborationofthese

nextarticleisconcerned.

eConceptsofComplex

e

madeuse,amongother things,ofthefact

cyspofthesourcemayassumevaluesin the

is.It waspointedoutthatsuchcomplex

urcesthatareexponentiallygrowingorde-

theformalprocedurerelatingtothe deter-

nseyieldsresultsthatapplyunalteredwhether

xspvaluesareultimatelysubstituted.

erbrieflytheresponseof theRLcircuit

whichwerewrite belowintheform

fthe impedanceZ(sp)=L(sp— Si)letus

nofFig.2,permittingsp thefreedomofthe

g.7.With

plainto

theRL

mplex

real

ample,

omesa

particular

mped

opposite

esult as

remains

facthas

ncharac-

orthe

elessourattitudetowardand ourreasoning

econceptbecomesmoreflexibleasa result

quencysptobe unrestricted.

nenttothe

Fig.1 when

mpedsinu-

)


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



LEXFREQUENCYANDIMPEDANCE413

e,thattheexpression38i scompletely

othetwovaluessp andSi;interchanging

nchanged.Hencewecouldjustaswell

equencyandspasthenaturalfrequencyof

e resultisthesamewhetherthe source

nthe circuitorthecircuitmorehighlydamped

xis andthenregardSias definingthe

circuit,wenolongerhavethesamecircuit.

inEq.38by addingtotheexpressionwithin

weseethatsp anditsconjugateareinvolved

rcuitwithapairof conjugateimaginary

encies.Thisarrangementofnaturalfre-

ediscussioninArt.5, Ch.5,characterizes

herefore,thatEq. 38canrepresentthe

xcitedbya dampedexponentialvoltageas

sponseofan RLcircuittoanappliedsinus-

mpedorundamped.

one'smindby thissortofreasoningare

eideaoftrading circuitswithsources,soto

moreelaboratesituations,toa hostofinter-

examplesassimpleas theRLcircuit,the

retativethinkingallowsustorecognizethat

tphysicalsituationsliketheLCcircuitexcited

andtheRLcircuitexcitedby asteadysinusoid

the skin"sinceonesimpleexpression(like

viorofboth.The economyandcircumspec-

tsmoregeneralexploitationportendarein

een toexistwithregardto Eq.32repre-

RLCcircuitto thesuddenapplicationofa

requencysp.Heretheexpressionwithinthe

zedasbeingcompletelysymmetricalwith

tiessp,Si, s2.Withspreal,the firsttermin

ealandthe lasttwoareconjugatecomplex.

inthesquarebracketsis thenreal,andthe

ce.Ifwenowregardspasdefininga simple

Lcircuit,andtheconjugatessi ands2as

dalsource,weobtainavalid physicalinter-

ipexpressedbyEq.32 thatisotherthan

rivation,andagainthetradingofa source

hesemattersare nowexpressedmorepre-

aldetaileddiscussionwhichfollows.


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e




mains

nthisarticleare thosenormallyassociated

cetransformmethodstobediscussedlater

ispoint hasatwofoldobjective,namely,

umspectandflexibleviewpointmayjustas

inedthroughtheclassicalapproachinvolving

thedifferentialequations,andthus,in the

omakepossibletheultimateachievementof

nderstandingoftransformmethods.

tetheresponseoftheRL circuittoan

plexfrequencysp(asgivenbyEq.10) inthe

9)

dAiare expressibleas

enlettings =sp,weget

formula40gives

titutionof41and 42into39yieldsthe result

oftheRLcircuitto anappliedvoltage

s

r Akintheform

sk),whichisin thisformulamerelytocancel

sp)or(s— Si)accordingtowhetherk=por

sistsoftheproductoftwothings,namely:

E/(s -sp)(45)


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



OMAINS

asmentionedabove,istheadmittanceofthe

e

hecomplexvariables,thisadmittanceis

hecircuitinthe so-calledsdomainorfrequency

ntities45 isanalogouslyspokenofasa

ceinthefrequencydomain,sinceiti sthe

ndsp.As afunctionoftime,thissourceis

n46fors= sp,whichisreferredtoas arepre-

he timedomain.Thesourceisthusregarded

ons,namely,oneinthetimedomain(this

n)andone inthefrequencydomain,which

Onecanformulatebyinspectionasimple

requencyfunctionE(s)intothetimefunc-

(47)

ocesstoF(s)yields

48)

ngtheunit voltageimpulseresponseofthe

erecall,itsnaturalbehavior.Thatis to

atrepresentshow,followinganinitialshock,

eftaloneandis, therefore,afunctionthat

selfinthetimedomain.

Eqs.44 and45,wenowcansaythat the

resents,in thefrequencydomain,the

e appliedvoltagee,(t),sinceEq.44together

sformationofthisresponsefromthefre-

accordingtoa logicalextensionofthepat-

8.

resultingbracketexpressioninEq.39

thatthe desiredi(t)isnot theresponseto

oltagee,(t)in47,butrather itistheresponse

e realpartof47. Asexplainedearlier(Art.

kingtherealpartcanbe postponeduntil

s.


/ h t t p : / / w w w . h





heusefulconceptthatall threequantities

hincircuittheory,namely,thecircuit,the

se,haverepresentationsbothasfunctionsof

ydomain)andasfunctionsoft(that is,in

wehaveestablishedthefactthatthe re-

omainisfoundsimplythrough multiplying

onsforthesourceandthe circuitinthat

verecognizedapatternforconvertinga fre-

toatime-domainfunction.Althoughthe

oesnotbecomestronglyapparentuntil one

laboratenetworkproblems,itiswell tomake

liminarywaywhilestilldiscussingthe

eexamples.

pplicationsoftheseideas,it ispreferable

g toconsidertakingtherealpart ofthe

ecureforthis difficultyishadthrough

ontainedwithinthesquarebracketsin Eq.

foranappliedvoltagethat istheconjugate

s, avoltage

s adesignationfortheconjugatecomplex

um(thebracketexpressioninEq.39 plus

theconjugateexcitation)isequalto the

ultofthesemanipulationsintheform

ghuseofthesuperpositionprinciple)that

A* arenowgivenby


/ h t t p : / / w w w . h




MAINS

eexcitationis

sp(54)

quencydomainmustbe thatpartinsidethe

an(s— S*)K(s),prefixedbythefactor1/2;

sp= jupsothats4= — jup, thenEq. 54

= 0,andthefrequencyfunction55is

)

egativeimaginary(E= — j\E|)and

s

nspiteofSo —* °°)

59)

msthatexpression54for theappliedvoltage

erally(withEandsp complex)itiscapableof

usoidwithanyfrequency,dampingconstant,

Equations50and51or 52givethenet

tosuchanappliedvoltage.

,insteadofEq.51or 52,

60)

s. 53and55,andY(s) istheadmittance

43.Whatwassaidearlierabouttrading

owberestatedmorespecificallythrough

n expression60,confusethefunctions

part.So longastheirproductdoes not

lt50 changes.

erminEq.50 representsthetransientpart

ntaryfunction),whiletheothertwoterms

ntegralor "steadystate."Ifweinter-

)and B(s),thatisto say,letE(s),Eq.55,

cuitandregardY(s),Eq.43, asthefrequency-


/ h t t p : / / w w w . h





avoltageexcitation(itstime-domainrepre-

), thenEq.50correctlyyieldsthenet

term isaparticularintegralandthe last

mentaryfunctionortransientpartof the

reoftheimpliedcircuitis notimmediately

toknowofa circuitthathasanadmittance

nalsthatmatchestheformofexpression55.

earemanycircuitsthatcanhavean admit-

Withs0 =0wesee fromEq.129ofCh.6

nceoftheseriesRLCcircuit.Fora pure

omestheadmittanceofanLCcircuit.Equa-

tobecapableofrepresentingthenetresponse

Ccircuittoan appliedvoltagethatisa

theoneinEq.48.

eawareofthefactthat thetimefunc-

ussionsarepertinentonly totheinterval

hemomenttheexcitationisapplied)which

at/ =0.Thus,thetime functionsinvolved

plyingonly fort>0. Fort<0 thecircuit

alltimefunctionsareidenticallyzerothrough-

enwesay,for example,thatE(s)=

-domainrepresentation(calledthetrans-

xpectittobetacitlyunderstoodthat e,(l)

ualto thestatedexponentialfunctiononly

entjustmadeisstill true,becauseitis

ue,andhenceit mustbetruefor azero

functionisonethat iszerofort <0,and

t>0.This isthedescriptionofa step

weseethatitstransformis E/s.Thatis

thatE/sisthe frequency-domainrepresen-

.47,describingtheprocessof transforma-

thetime domain,wenowobservethatthis

fwedifferentiateitwith respecttothetime.

ebracketthus becomesmultipliedbys,we

hatdifferentiationinthe timedomaincorre-

ysin thefrequencydomain;and,sinceintegra-

n thetimedomain,itmustcorrespondin

visionbys,as maybeverifiedthroughinte-

q.47.


/ h t t p : / / w w w . h




ELUMPED-CONSTANTNETWORK419

ulseu0(l)is thederivativeoftheunit

hatthe transformofu0(<)mustbetheconstant

fu—i(0has justbeenshowntobe l/s.The

arityfunctionsinthefamilyun(t) arethus

ns50and60, wecannowinjectevengreater

etation.Thuswemay,forexample,regard

60 asanewadmittanceY'(s)andasso-

s)=1so thatY(s).E(s)=Y'(s).E'(s).

mefunction,referredtoastheinversetrans-

ulse,Eq.50nowis interpretedastheimpulse

gtheadmittancefunction(Eqs.43and 55)

uss theproblemofhowacircuithavingthis

but,whateverphysicalstructureitmay

uralbehavioris describedbyEq.50.That

equilibriumequationofthisnetwork,Eq.50

ementaryfunction;allthreequantitiessi,

requencies.

ulwaysofinterpretinga givenexpression

ponseareseentosuggestthemselvesquite

tablishedtheideaofthe frequency-and

equantitieswehavetodealwith,together

Eq. 60totheeffectthatthe responsein

eproductofthe frequency-domainfunctions

andtheexcitation.Asmentionedabove,

(s)]are spokenofasthetransformsofthe

,andthelatterasthei nversetransformsofthe

productofY(s)andE(s)is thetransform

dthe correspondinginversetransform,or

oughthesimpleprocessmadeevidentby

owingarticlethesemattersaregeneralized

ped-parametercircuits.

orAnyFiniteLumped-Constant

eneralizetheprecedingdiscussiontoshow

edureappliesnomatterwhat thenetwork

ughinthesediscussionsweshallassumethe


/ h t t p : / / w w w . h





ationandthecurrenti(t)a response,the

tiallyunalterediftherolesofe(t) andi(t)are

tionofthedifferentialequationexpressing

must,ofcourse,takeintoaccountwhichof

theexcitationandwhichisthedesired

uationingeneraldoes notapplytoboth

olvedinthesemattersare suppliedinthe

momentweareconcernedonlywiththe

erentialequation.

nhas theform

Veit)(62)

with,the followingsimplerformwhichthis

opriatecircumstancestake,andlatershow

procedureareneededtoaccommodatethe

(t)=e(t)(63)

versionofEq.62involvesno derivative

).Choosingthecoefficientan=1 evidently

ughtforthespecifications

oneassumes

)

nintoEq.63leads,in anowfamiliarmanner,

Oo

mial

••+aiS+ao(67)


/ h t t p : / / w w w . h





sn,result 66maybeexpressedinthe form

Sn)

to thecorrespondinghomogeneousform

quilibriumequation)

= 0

ndcancelationofthefactore"yields

+ao)B= P(s)-B=0(71)

ution(B^0) leadstothecharacteristic

stic valuesSi,«2,'",Sn,whichare also

uralfrequenciesofthecir cuit.Sincean

eformgiveninEq. 70satisfiesEq.69for s

sofEq.72, themostgeneralcomplementary

+Bne'•i(73)

edifferentialEq.63 isthusgivenby

B2e'«+•..+Bne''1+ Bpe'* (74)

mEq.66. TheB\••• Bnareintegration

erminedfroma knowledgeofthestateof

eterminationisreadilyaccomplishedin

eexcitationfunctionmadein Eq.64we

nuousat t=0.That istosay,at theinitial

roto thevalueE.Hencethevalue ofthe

st atthisinstantlikewisejumpfrom zero

hatthisjumpis restrictedtotheterm

ivativewerediscontinuousatt =0,then

dhaveinfinitevalues,andthe requiredfinite

ult.Sincei{t) =0fort <0(thecircuit is


/ h t t p : / / w w w . h





e/ =0),thecontinuityrequirementjust

t=0(75)

einstantimmediatelyafterswitchclosure.

nditionssufficetodeterminethen inte-

malsolution74,for theyyieldtheequations

spBp

p(76)

1Bn=-

hconsideringEq.74andits n— 1successive

stemofequationshas numerousinteresting

udiesdueprincipallytoVandermondeand

etowritethedesiredsolutionsfor theBi•• •Bn

inantreads

theory,itsvalueisunchangediftheelements

eplacedbythe differencesbetweenthen.

he respectiveelementsofthefirstcolumn.

as theform78fromwhichall oftheele-

••S»—1


/ h t t p : / / w w w . h





olumnareseento containthefactor(s2— Si),

thisfactor.

edversionofDformedthroughreplacing

columnin77bythe differencesbetweentheir

he respectiveelementsofthefirstcolumn,

mustalsocontainthefactor(s3— s{).In

oncludethatDmustcontainthe factors

,(sn— Si);and,throughformingnew3d,4th,

ngtherespectiveelementsofthesecond,one

,(s4— s2),••. ,(sn— s2)mustbefactorsofD.

mmaryofthesethoughtsleadstothe result

(s,.— sj)forall * >j=1, 2,•••, n— 1

cethe totalnumberofthesefactorsis

n(n— l)/2,and,sincethetermsin

ationofD(obtained,forexample,through

dure)arebyinspectionseentobe homo-

1 +2+3H h(n— 1)=n(n— l)/2

valueofDcandifferfromthe productofall

aconstantmultiplier.Thatisto say,the

xpressibleas

ndalltherowsof factorsaretobemulti-

rmofexpression79 itisevident

inedintheprocessofits evaluationreads

pectionofform77 shows,ontheotherhand,

tsonthe principaldiagonal,whichisaterm

s2s32s43••• s„n—1.Itfollowsthatk =1,

eresult

1, 2, ••• , n-1


/ h t t p : / / w w w . h





onfor anyunknownBkintheEq.set

measin 77exceptthattheelements1,

h column.Thedeterminantinvolvedhere

yin thatthequantitysptakesthe placeof

xpressionfor Bkallfactors(s,— sj)cancel

atorforwhicht =porj =p,andthosein

i=k orj= fc.Hence(asmaybestbeseen

q.81yields

s„-s*—i)(s*+1-sp)•••(sn-sp)

-S*—i)(si+i-sk). ..(sn- sk)

(s,.— sj)startingwitht =Jb+ 1as

ebecausethenumberof suchfactorsisthe

nominator,referencetoEq.68for Bpshows

82willcausecancelationofallthe numer-

enominatorfactorsin68except(sp— sk).

k— sp)andcancelingtheminussignwith

k-i)(sk-Sjfc+i).••(sk-sn)(sk- Sp)

67written inthefactoredform

s-sn)(84)

givenby Eq.83inthe morecompactform

)

ontainalso theresultshowninEq.68.

hecompletesolutiongivenbyEq.74are


/ h t t p : / / w w w . h





=1, 2,narethose associatedwith

ioni0(t)andthatfork= pyieldstheampli-

gral.

raldifferentialequationasgivenby Eq.62

ndsideas thesumoftheexcitationvoltages

utionisgivenby addingtogether

erivativeofEq.74

dderivativeofEq. 74

erivativeof Eq.74

esultingsolutionto Eq.62as

An<* +AJ* (87)

+hS+b0 (89)

, p)(91)

calinformwithEq.44 pertainingtotheRL

einthepresentgeneralcaseis thatthe

givenbyEq.90insteadofby Eq.43,has

zedform.

nsarepertinent tothecomplexconstituent

givenbyEq.64, theactualsolution(asin

n bytherealpartof thecomplexform87.

sityforindicatingthisstep, justaswedid in

ousarticlepertainingto theRLcircuit,

hecorrespondingresultobtainedforthe


/ h t t p : / / w w w . h




STEADY.STATEANDTRANSIENT

geexcitation.Thisprocessleadstothe final

ne"1+VV+ (92)

87 exceptthatatermwiths, =spis added,

Akare nolongergivenbyEq.91. Instead,

xpressibleasindicatedinEq. 51pertinentto

cuit.Theend resultissummarizedby

owingformula(like Eq.60)forthecon-

-1, 2, . •n, p, q(93)

excitationvoltageinthetimedomain

with theanalogousonesdiscussedforthe

furtherelaborated.

ousarticleabouttime-domainandfre-

tationsofboththesourceand thenetwork,as

,inwholeorinpart,of thefunctionsY(s)

teinterpretationoftheresultsisthus seen

o themostgenerallumpedlinearnetwork.

iumEquationsfor Driving-Point

eciprocityAgain

pbetweentheadmittancefunctionY(s)

ialequation,asmadeevidentbyinspection

itis seenthatonemayderivethis differential

ctingtheappropriateadmittanceorimped-

efordoingthisfollowsthe samepatternas

nputortransferrelationfora purelyresistive

hatthedesireddifferentialequationcanin

utrecoursetomeshor nodeequationsand

onprocedures.Anillustrativeexamplewill

meantbytheseremarks.


/ h t t p : / / w w w . h




POINTANDTRANSFERSITUATIONS427

sthe complexamplitudeofavoltage

mplexamplitudesofinputandoutputcur-

numericalelementvaluesareinohms,

oblemisto writedifferentialequilibrium

ationei(t)=Eie'1with responsecurrents

.

ifferentialEqs. 104and105arepertinent.

gthe impedanceconcept,wemaywrite

ssumingforthemomentthatI2= 1,

+6

thenetwork,thecorrespondingvoltages

valueof I2aregivenbythe aboveexpres-

ceonehas

+6

7s+6


/ h t t p : / / w w w . h





alequationsarenowrecognizedtobe

7-+6)ti(0

ei(<)(104)

-+6)i2(<=24— (105)

tionwiththeseresults,toobservethatboth

areresponsefunctions(ratiosofoutput

inwhichtherolesof excitationandresponseareinter-

edifferentialEq.110ratherthan105 nowapplies.

procalofthetransferadmittanceF12(S)in

y,the so-calleddriving-pointfunction

i,regardlessofwhichis thesourceand

g. 8wecanequallywellregardthe source

nd interpretE1asthe resultingterminal

rfunctionF12(s)representstherelationship

Eiis theexcitationand/2theresponse,as

onofthisfunction.Correspondingly,we

.104to describetheequilibriumconditions

workwitheithere^t)or t'i(t)asthesource,

eindicatedtransferrelationshiponlywith

2(t)asan excitationandei(t)asaresponse,

he redrawnversionofFig.8as shownin

mportantrespects.Firsti nordertopermit

putterminalsmustbeopen-circuited,and

ceJ2 isasource,Ebis nolongersimply

roughthe6-faradcapacitanceinserieswith

tisthealgebraicsumof thispassivevoltage


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



POINTANDTRANSFERSITUATIONS429

fthe source(Eq.97isno longerapplicable).

voltageisnot known,wecannotproceedin

e.

ttheotherend ofthenetworkandassume

a,we thenhave

107)

reEitimesas great.Hence

lartothereciprocalofF12(s) inEq.103,

tementthatthereciprocalofa transfer

sferfunction.Thetransferimpedance109

theexcitation,whilethetransferadmit-

enE^is theexcitation.

ppropriatetothesituationdepictedin

tobe

(t) (HO)

aysregardedasthe ratioofresponseto

ut.Thus,multiplicationofaresponsefunc-

nyieldsthe associatedresponse.Whenboth

sponse)pertaintothe samepointinthenet-

asadriving-pointfunction[likeFn(s)or

e responseandexcitationareatdifferent

likeF12(s)in Eq.103orZ12in Eq.109]is

ybeenpointedout,thelatteris invariant

nts ofexcitationandobservation,astate-

citytheorem.Itsproofin termsoftheimped-

amepatternasthatgivenin Art.6,Ch.3,


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e




htswemaysaythatthe ratioofIito

enticalwiththe ratioofI2to Ei(Eq.103l

ratioofE2to Iiinthe circuitofFig.11is

toI2(Eq.109) inFig.9.In comparing

8and10, itisworthnoting thatonecould

withthepointsofexcitationandobservationinter-

ciprocitytheorem),showingthatthedifferentialEq.105

changed)doesapplytothissituation.

bothvoltagesourcesinthesame circuit.

itcharacterofavoltagesource,thesituation

2=0,andthatinF ig. 10forEi=0. Ina

mbineFigs.9and11, beingmindfulofthe

acurrentsource.

0 withtherolesofexcitationandresponseinterchanged,

ointsofexcitationandobservationinterchanged.Bythe

ifferentialEq.110(withletterseand tinterchanged)does

ncefunction,thepointsofexcitationand

-circuited,whileforatransferadmittance

rt-circuited.Ineithercase,thepertinent

-or short-circuitcharacter,sothatits transfer

theotherleavesthecircuitundisturbed.

atisessentialinanyspecificapplicationofthe

ctionalsotopointoutthat,if, foragiven

(open-circuit)driving-pointortransferim-


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



ERALSOLUTION

geresponseacrossanynodepair tocurrent

meorat anyothernodepair),thenall such

sthavethesamepolesbecausethesearethe

givennetwork.Thatisto say,theyarethe

llationresultingfromadisturbancesuchas

ntothenetworkthroughtheapplicationofa

depair,or thetossingofchargesatthe capaci-

n.Nomatterwherethe excitationisapplied

ageresponse,thefrequenciesanddecrements

samebecausetheycharacterizethenatural

orkunderforce-freeconditions.

menetworkisavoltagesourceinseries

ponseisacurrentin thesamebranchorin

ratioof responsetoexcitationisa(short-

nsferadmittance;andallsuchfunctions

haremoreoverthesameas thepolesofthe

bovesincetheyareagainthenatural fre-

orkunderthesameconditions.Partialevi-

ybeseenin thefunctionsgivenbyEqs.102

sefactsgreatlysimplifiesthesolutionto

see,forexample,Prob.4attheend ofthis

onoftheseideasisgivenin Art.8.

Solution

ompletesolutionderivedinArt.4,and point

thatarefoundtobe usefulindealingwith

Inthisdiscussionitis assumedthatthe

ther esponseacurrent.Theresponsefunc-

g-pointortransferadmittanceY(s),Eq.90,

ninthefrequencydomainisthe voltage

d96.If,instead,theexcitationis acurrent

e,thenthe pertinentresponsefunctionbe-

,andtheexcitationinthefrequencydomain

ceptfor thisalmosttrivialchangein

lutionand thepertinentequationsyielding

are preciselythesame.

bewrittenin awaythatisfree fromany

atureof eithertheexcitationortheresponse,

rfectlyflexiblewithregardtothe proportion

thenetworksharein determiningthere-

nctionY(s).E(s)orZ(s).I(s)is inanycase

nthe frequencyvariables,wedenotethis


/ h t t p : / / w w w . h




ERALSOLUTION

unit stepu—i(t)it is1/s.Forthetime function

ewritten

sdeterminedinArt.3)is

writtenasinEqs. 95and96.

)pertinentto anyspecificexampleisa

nditsconversiontothetimefunction113

4is equallysimpleinformalthoughinsome

ationally.Applicationofthesecompact

ngexamplesismadein thenextarticle.Before

ms,however,itisveryhelpfultobe awareof

olutionwhichweare nowinapositionto

hagaintocallattentionto thefact(men-

rentiationinthetimedomaincorresponds

efrequencydomain,andtimeintegration

s.Morespecifically,

edbysF(s)

edby-F(s)

ntsis obviousfromEqs.113and114.Thus,

4bysF(s),it followsthatAkbecomesre-

.113this changeresultsfromdifferentiation


/ h t t p : / / w w w . h





therhand,ifF(s)in Eq.114isreplacedby

elds— ,whichresultisobtainedin thetime-

3throughindefiniteintegrationwithrespect

cationofthetimefunctionisthat accom-

oftheindependentvariabletbyt — l0where

angeamountstoarevisionof thetimeorigin,

etimefunctionasawhole.Thus,for aposi-

delayedbytoseconds.Ifthischangeis

otethat eachconstantAkbecomesreplaced

sinformula114if F(s)istherereplacedby

vethatmultiplicationinthefrequency

nttoa delayofthetimefunctionbyseconds.

o)

edbye°F(s)

mentarytothisoneisthatof replacingthe

— s0,whichcorrespondstoadisplacementof

ane.Themeaningofsucha displacement

derstoodifweconsiderthe effectofthechange

alfrequencyfactor(s— Sk),whichmaybe

sentingapole ofF(s)throughbeingafactor

omialD(s)asin Eq.112,butalsoas beinga

eratorpolynomialN(s)andhencerepre-

cesuch afrequencyfactorisconverted

earthatdisplacementofthefunctionF(s)

erpretedasa displacementofallthecritical

es)ofF(s)by thesameamounts0.Wesay

onstellationcharacterizingF(s)istranslated

0.

nalgebraicsignin thestatementthatif

»(s* +s0);thereasonis obviousfromthe

sothatsocanhaveany complexvalue.Spe-

splacementofthepole-zeroconstellationin

ntaldirection(paralleltothe realaxis);if

ementisin theverticaldirection(parallel

nctionis immediatelyevidentfromEq.

s* bysk+s0,theentirefunctionbecomes

wehavethestatement:


/ h t t p : / / w w w . h




ERALSOLUTION

s0)

edbye-f(t)

isresultis indicatedbythefollowing:

0)]

nofF(s— juo)isthat ofF(s)displacedin

by anamountuo,whiletheconstellation

displacednegativelyinthe jaxisdirection

mainfunctionwithinthebracketshasthe

thesedisplacedconstellations(asisobvious

oftwofunctionsisinfinitewhereeither one

rilyzerowhereeitheroneiszero).Usually

re notofinterestsincethe evaluationofthe

requiresonlya knowledgeofthepoles.

nctionin124 isageneralexpressionforan

erwave,thisparticularstatementhasmany

alizedbystatement123itis asimple

omplementarytothepropertyexpressedin

hatis tosay,wemayreadilydeterminethe

onofdifferentiatingorintegratingthe fre-

s,ifwedenoteasmallincrementin sby

hestatement123that

Ast)f(t)

t

nedin125becomesexactasAsisallowedto

er,whileF*(s)asgivenby Eq.12Gapproaches

espectto s.Sincethisprocesscanobvi-

enumberof timesinaforwardor reverse


/ h t t p : / / w w w . h





ltthat

dsn

(0

aluesfornimplyrepeatedindefiniteintegra-

ementaryprinciples,cancelsa likenumber

ns).

sedthe processoffrequencyscaling.It

iderthis topicagaininthe lightofourpres-

.Thus,ifin thefunctionF(s)weintroduce

catedbyletting s=a\with aequaltoa

uencyfunctionF(S)becomesreplacedby

esaresubjectto thesametransformation,

X* arethecriticalfrequenciesofF(X).The

)

a)and F(X)haveidenticalvaluesfors= aX

X),acomparisonofEqs.114 and129shows

),relatedto F(X)inthesamemannerthat

enby

=-/(-)= -f(t)(131)

resultsinthe statement:

akethechangeofvariable

ew frequencyfunctionF(\),

ecomesreplacedby /(t)=

ons=aXcorrespondstoa magnification

onofthefrequencyfunction.Thatis to

he samevaluesasF(s)exceptthatthey

meslarger.The frequency-domainfunc-

r(1/a).Sincethecorrespondingtransforma-

riableinthe timedomainreadst=r/a, we

iscompressedintheabscissadirection.Cor-

mefunctionoccur (1/a)timessooner.We


/ h t t p : / / w w w . h




ERALSOLUTION

yscaleiscontractedandthe timescaleex-

r(1/a).Meanwhiletheordinatesofthe time

esameratio,so thattheneteffectupon

tallerandshorter.Scalingthefrequency

ectuponboththeindependentandthedepend-

omain,andtheseeffectsaremutuallyinverse.

vingcomparablepracticalvalueisobtained

ng.Whentheexcitationfunctionapplied

nitimpulse,thenF(s)is identicalwiththe

mple,withthetransferimpedanceZi2(s)of

nthatthe transformoftheunitimpulseis

stepexcitationappliedtothissamenet-

ponseisclearlyequaltothevalue of212(0)

atelyactslikea directcurrentandthezero-

nsferfunctiondeterminesthed-cresponse.

epresponseisthe integraloftheunitim-

tevalueoftheunit stepresponseequalsthe

mpulseresponsecurve.Henceweseethat

ween/(<)andF(s)is independentofhow

s).Y(s)isapportionedbetweenthe

eexcitationandthenetworkresponsefunc-

texpressedbyEq.133is generallytrue,

sactuallyan impulseresponseorinsteadis

bitraryexcitation.Thetotalareaenclosed

sthezero-frequencyvalueofits transform.

notfinite,thevalueof F(0)willlikewisenot

ecomesmeaningless.SincethevalueofF(0)

eofvariables =a\(witha finite),itisclear

fectthenetarea enclosedbyf(t),whencea

hisfunctionmustbeaccompaniedbya

pansionorviceversa.Theresultfoundthrough

efrequencyscaleisthus seentobein agree-

byEq.133.

ycontainedinstatement128,we can


/ h t t p : / / w w w . h




STEADY-STATEANDTRANSiem

ngfulonlyifthenth derivativeofF(»)'

.133 hasaninterestingimplicatior

acteroff(i)mustbe suchthatthenet,s

oniszero.If /(<)hasadampedoscilla.1

sechangesign atleastonce),thenthea

dnegativehumpsmustcancel.* ^

ooforderkat s=0(that is,ifitcont

oregeneralrelation134that

2, k -1

somecasestodrawusefulconclusionsrej

terof/(/),whichisthusseen tobere

s=0.

onship,namely,thatlinkingtheas;

hecharacterof /(/)fort= 0,mayaL

tiongivenabove.If wewriteEq.11

m

ao

4 and5showsthatwemay regard/

tialequation

tionmo(0isthefamiliarunit impulse,

alequation

=u-,(<)

ichthisresultdoes notnecessarilyfollowoccuj

),accordingtoEq.113,hastermsthat donotdeej

/(<)-cosat forwhichEq.57showsthat

O)-0,thenet areaunder/(<)isnotdefinsble.


/ h t t p : / / w w w . h




ERALSOLUTION

ctionistheunit step,thenitis clearthat

«)+•••+&m/o(m+1)(0(140)

kthderivativeoffo(t).

ediscontinuityintheright-handmember

ythehighest derivativetermontheleft,

msremaincontinuousatthis instant,where-

=0(141)

6I/o(*+2)(0)+•• •+6m/0(i+m+I)(0)(142)

-1(143)

. , v -n-m -2(144)

viorofF(s)accordingtoEq.137is de-

ppearsasl/s'+2,thenwe canimmediately

dtimefunctionf(t)andits successivederiva-

arezerofort =0;thefirst nonzeroinitial

+1.Forthesingularityfunctions,f(t)=

at»= — (n+2).Theunit linearramp

husyieldsp =0;theunit parabolicramp

— 1,andsoforth.Theseresultsareobvi-

atement144.

successivederivativesarezerofort =0,

efunctionis ratherslowingettingunder

mthisfactmeansthat thetransientresponse

propertystatedbyrelations144 and145

eresponseobviousfromaninspectionof the


/ h t t p : / / w w w . h





hall considerisillustratedbythe sequence

2for whichthetransferfunctionisthe

2/h=l/(«+1)" (146)

tively.* Thesenetworksarereadilyfound

orkshavingthe transferimpedance146forinteger

h5.

pedance^^(s)bya methodofsynthesisto

eader maychecktheresult146in each

reusedmerelyasa designationfortheimpedance

rder.


/ h t t p : / / w w w . h




ohaveannth-orderpoleat thepoints= — 1

thecomplexfrequencyplane.Letour

he unitimpulseresponseofthisseries of

sumingtheinput currenttobet'i =«o(<),

tputvoltagee2(t) fort>0.

excitationinthis caseisunity,wehave

ththesimplestnetworkforwhichn =1,

nce(orbythe methodofthischapter)the

0(147)

Zi2m/ds(148)

8immediatelytellsus thatthetimefunction

ads

<e~'fort> 0(149)

e timefunctioncorrespondington=3,

entthatthe impulseresponseofnthorder

kstheresultsgiven byEqs.147,149,and

ervalueofnyieldsat oncetheimpulseresponse

ondingorder.

wewish toknowtheunitstep response,

e timefunctionsjustfound.Throughinte-

gwithf2(t)and proceedinginsequence,itis

itstepresponseof ordernisgiven by

h(t)+...+/n(<)](153)


/ h t t p : / / w w w . h





ethatthebracketexpressionequalsthe

rinexpansionofe'.Throughoutthetime

these termsrepresentareasonableapproxi-

mese-' isapproximatelyequaltounity,

eyondthisintervalthebrackettimese~i

(J)tendstowardunity whichisitsultimate

thusseentobe delayedandstretchedinto

ofnetworkwiththeresponsefunction146.

eferredtoasthe"precursor"interval

issubstantiallyzero,and theoneimmediately

eresponsegraduallyrises,isthe so-called

on."Thereaftertheresponseisequalessen-

e.

52fortheimpulseresponse,we observe

ofasinglehumpwitha maximumequalto

156)

eisthe derivativeofthestepresponse,we

presentsthemaximumslopeofthestepre-

the timeofitsoccurrence,whichisroughly

alofsignalformation.Thismostimportant

seis thusseentobe delayedintimepro-

epoleinthe responsefunction146.

delaytime asmuchaswewishthrough

ofthefrequencyscaleaccordingtostatement

welet

omes

sgiven by


/ h t t p : / / w w w . h




ularfrequencyscalinggivenby Eq.157

ationofourtimescale,sincethetimeof signal

second,independentoftheordern. Since

tion157becomesdegenerateforn=1,these

ordersn=2, 3,

erimpedanceischangedfromthevalue

sforn valuesupto5are thosegiveninFig.

capacitancevaluesdividedby(n— 1),due

valueappropriatetoeachorder.In this

network(whichisthesameasbefore)is char-

ctionhavingadoublepole atX=— 1;forthe

unctionhasathird-order poleatX=—2;

unctionhas afourth-orderpoleatX= —3,

esponseforthissequenceofnetworksare

dersn= 2,3,4,5,while Fig.14showsthe

sponse(Eq. 159)ofthenetworksinFig.12 modifiedby

esdividedby(n— 1).


/ h t t p : / / w w w . h




ponseforn =5only.Evenforthis order

unction.Networksofthistypearecharac-

nsewhichstemsfrom thefactthatthenatural

onseofthenetworkinFig.12(e)with itsLandC

atory.Weshallnowconsidersomecases

ngerapplies.

15has thetransferimpedance

tter-

ansfer

163.

agnitudeofthistransferfunctionfor

tesunityovertherange— 1<u <1and

workisaconstituentofa classofso-called

heresponsecharacteristic

sequenceofnetworkshavingthisresponse

rworthfilters"(afterthechapwhofirstused

pose)and nisreferredto astheorderofa

ype.Thusthecircuitof Fig.15isreferred


/ h t t p : / / w w w . h




rworthfilter,theoneof firstorderbeing

hesequencedefinedbyEq.146andhence

stthreefunctionsofthis type.Theyall

(the half-powerpoint)atu=1, and

esponsecharacteristics(Eq.167)forButterworthfilters

yieldabetterapproximationtotherectangle

eideallow-passfilter characteristic.

mpulseresponseof thisnetwork,then

fiedwiththefrequencyfunctionF(s)inEq.

t

Si)(168)

ticalfrequenciesasgiveninEq.165we have

efunction

5=V2e~t/V~2sin(t/V2)(170)

unitstepresponse,wemayintegratethe

dorconsiderthe appropriatelymodified

hetransformoftheexcitationnowis

1)

avethe criticalfrequencys0=0inaddi-


/ h t t p : / / w w w . h





165.UseofEq.114 thenyields

aicallysimpleexpressions,itisnevertheless

etricalinterpretationinthes plane,since

f(a)thepoles ofthefunction171;and(b), (c),and(d),

eexpressions172forthe coefficientsAo,A\,Atrespec-

tionofthestepresponseof thesecond-orderButterworth

mesextremelyhelpfulinmoreelaboratesitua-

esketchesinFig. 17,part(a)is aportion

smarkingthepositionsofthepoles s0,Si,s2

171.Inparts(b),(c), (d)areshownrespec-

ationsforthefrequencyfactors(S,— «,)

ons172forA0, A\,A2.Fromthesketchof


/ h t t p : / / w w w . h




ionthattheanglesofthe twovectorscancel,

h isunitywehaveat oncethatAo=1,a

sily seenfromthegraphicalportrayalof

nitisobtainablethroughtheir numerical

entalgebraicexpression.Infact,onecan

A'salmostataglance.Thusfromthe

hatthenetangleequals 90° plus135°,and

agnitudesofthetwofactorsisy/2;so

ingtothesketchofpart(d) obviously

epresentingthefrequencyfactorsforA0

eforAihavetheir tipsatSi,and thosefor

eachcasethefactorscollectivelyforma

ngfromthevariousothercriticalfrequencies,

particularcriticalfrequencypertinenttothe

ay,theyallconvergeuponso forthedeter-

thedeterminationofAi,andsoforth.

givenbythereciprocalproductofthe perti-

atthis simplegeometricalpictureapplies

oregeneralsituationstheexpressionsfor

atoraswellas denominatorfactors.

eunit stepresponseofthenetworkof

=-e^'Yv^(173)

timefunctionaccordingtoEq.113becomes

-(t/4)](174)

resultthroughintegrationof Eq.170.He

Eq.170throughnotingthatthepresentprob-

eriesRLCcircuitinFig.21of Ch.5,and,

the circuitequalsthevoltageacrossR.

uldinessenceagreewithEq.170 above.

problemwehaveC=.y/2,ud =a=1/.\/2>

verifiedifwerecallthat theexcitationinthe

ntimpulseofvalueC.


/ h t t p : / / w w w . h





onseaccordingtoEq.174is shownin

e intervalofsignalformationsetsinalmost

rastwiththeresponseshowninFig.14, the

of thesecond-orderButterworthfiltershowninFig.15.

eultimatevalueunityandapproachesitin

aximumovershootofabout4percent occur-

ommontransferimpedance(apartfromafactor1/2)is

hfunctiongiveninEqs.175and 176respectivelyforthe

ndthesingle-loadedcircuit(b).Elementvaluesarein

thecircuitsofFig.19 whicharethird-order

efirstofthese [part(a)]thetransferimped-

] wehave


/ h t t p : / / w w w . h




rksthecurrentsourceinparallelwith the

nvertedintoanequivalentvoltageEi(numer-

withthisresistance.TheratioE2/Exis

husthefirst ofthesetwonetworksis

esistivevoltagesourcedrive,whilethesecond

currentsourcedrive.Bothhavethesame

third-orderButterworthfunction177.

orafactor1/2.Supposeweconsideronly

edformthisone reads

3=-1/2-;V3/2(178)

attern ofthistransferfunctioninthe

-orderButterworthfilter,thepolesaresym-

efthalfof theunitcircle.

nitstepresponse.Thenweagain have

itation7i(s) =1/sand

)

contributedbythesourcefunction.Equa-


/ h t t p : / / w w w . h




thesecoefficientsaregeometricallyp

owninFig.21. Beingmindfulof

howingthefrequencyfactorsinthecoefficients

polesonthe unitcircle,onemay1

es

3=-jly/\


/ h t t p : / / w w w . h




enyields thedesiredtimefunction

-2

howninFig.22. Fromacomparisonwith

tthestep responseofthethird-orderButter-

slowerthanthatpertinenttothe filterof

third-orderButterworthfiltersshowninFig.19.

ximumfallsataboutt =5secondsascom-

s),buttheamountofoverswingandoscilla-

markedlylarger,thefirstmaximumbeing

nitywhilethesecond-orderfilterexhibitsa

mighthavepredictedthisgreatertendency

derfiltertoexhibitan oscillatoryresponse

circuitforwhichthesinusoidalsteady-stateresponseis

tepolesis relativelyclosertothej axis.

atthispair ofpoleslieson radiallinesthat

ereasinFig.17pertinenttothe second-order

lines.Thesecond-ordernetworkismore

rd-orderone.


/ h t t p : / / w w w . h





nsientresponseofthe pairofloosely

sesinusoidalsteady-stateresponseisevaluated

ofthiscircuit(Fig.28 ofCh.6)is redrawn

umptionthat(Ci/C)«1,the twoparallel

upled,andthemagnitudeoftheratio E2/Ii

cteristic(asshowninFig.32of Ch.6)that

redwiththecenterfrequencyoftheband

othe resonantfrequencyofeitherparallel

6, thetransferimpedanceofthiscircuit

n

2as+ub3)

hverynearlythesameasw0, butitis important

at

howsaportionof thesplanenearthe posi-

,thedifference185betweentheimaginary

circuitofFig.23with inductiveratherthancapacitive

denotedby 2a;anditis pointedoutinthe

hatthevalueof theparameterarelative

= 1/2RCoftheparallelRLCcircuitsis

hecharacteroftheresultantbandpasschar-

(a)and(b)ofFig. 32,Ch.6].

t"characteristicresultsifwechoose

iththisconditionimplied,comparisonof

neofthe sketchesinFig.17aboveforthe

hfilterofFig.15(ignoringthesourcepoleat


/ h t t p : / / w w w . h




evicinityoftheirpassbands,thesefilters

alpolepatterns.Wemaysaythatthe maxi-

fFig.23 hasasecond-orderButterworth

ssbandiscenteredatw= woinsteadofbeing

dentallyofpracticalinterestto notethat

thesameresponsecharacteristicusinginduc-

ecoupling,asisshownin thecircuitofFig.

oseif(L/L{)«1.The readermayshow

r thiscircuitweget

+ £062)

d185(theidentities ofwaandubare inter-

ignificantdifference).

uctivecouplingiscommonlyaccom-

hemagneticfieldsofthetwo self-inductances

ductivecouplinginFig.24 maybereplacedbyan

ance.

heinductivepartofthe circuitisthen

5 wherethetopsketchshowstheinduct-

e circuitofFig.24andthe bottomoneisits

utualinductanceM.Sincebyassumption

simplematterto establishtherelationship

thetopcircuitpracticallynoneofIi

etheverymuchsmallerL isinparallelwith

parallelL,therefore,isIiLs,and £2equals


/ h t t p : / / w w w . h





s value.Thatis,

,

oupledinductancesonehas

sought.Thus,ifin Fig.24wereplacethe

valentmutuallycoupledpairasshownin

ance186ischangedonlyi nthatthefactor

placed

mes

ersit

hatsome

the in-

hatis

willhave

withit.

circuits

onshown

n

this modifiedcircuithasessentiallythe

high(RCo>05^1)andso longas

esistance

ofFigs. 23

d(Eq. 192)

avior.

ociatedwiththecapacitances,asinFigs.23

emecase

ociatedwiththeinductances.Theresponse

snot affected(exceptforsecond-ordereffects)

eassociatedlossesso longastheenergylost

dwiththepeakvalue ofthestoredenergy

w-losscircuit;seeArt. 6ofCh.7).

itiona =aleadingto themaximallyflat

dwithacomputationofthetransientresponse


/ h t t p : / / w w w . h




uned"circuit.Sincea=1/2RCand,by

=Ciw0/C=Lu0/Li=Mw0/Laccording

acitivelyorinductivelycoupledcircuits,the

metthroughsetting

)

esforthesecircuitsaregivenrespectivelyby

4)

)

ninput

timpulse.

alwithZ12(s),

on196,the

s

vesketchof

shownin

efact that

tetheexpres-

uswesee thatSi«ju0, (st— s2)

trans-

elevant

tsof


/ h t t p : / / w w w . h




DY-STATEANDTRANSIENT

~j2u0.Henceweget

3yields thetimefunction

oneobtains thesameresultusingthetransfer

dermayeasilyverify.

rvethatthe function201mayberegardedasa

pidlyvaryingoscillationsinu^.Apartfrom

multiplier,thisenvelopefunctionisobserved

htheimpulseresponse170ofthe second-

veninFig.15, aresultthatstemsfromthe

t(fora =1/\/2)thepolepatternof our

tyofits resonanceregionisidenticalwith

erinFig.15 forthevicinityofits midband

egionwhichiscenteredats= 0.Wemay

sina sensethebandpassanalogueofthe

d thepropertyexpressedbythestatement

clusion.

esponseweevaluatetheunitstepresponse

ehave Ji(s)=1/s,theonly differenceis

onF(s)isdividedby s.Asmayreadilybe

the correspondingcoefficientsAiandA3

dedbyju0whileA2and A4turnouttobe

by— jwo.Thesechangesresultinyielding

>ot

mecharacterastheunitimpulseresponse201,

nvelopeis&>otimessmallerandthe enclosed

ftedby90°.


/ h t t p : / / w w w . h




S

ecedingexamplesshowthatthereisa very

acterbetweentheimpulseresponseandthe

passnetworkthetwoare essentiallyalike

erencein

lyappre-

rcumstance

rthelow-

yis itsmid-

thesudden

erofre-

nlyapply-

emidband

econsider

suddenap-

avingits

eshould

nalogousto

owpasscircuit.

excitation

cordingto

dance196we thenhave

func-

esponseof

Fig.23

204.

S4)(s- ju0)(s+ jwo)

oustothatin Fig.27isshownin Fig.28.

ffersonlyin thatthepolesat±So =±jwo

patternin thevicinityofSo thatislikethe

entto thestepresponseofthe circuitof

valuatesixcoefficientsAkusingformula114.

ftout.Visualizingthe frequencyfactorsas


/ h t t p : / / w w w . h





chofFig.28, wenoteforthecomputationof

4,(s0 -S3)=ay/2e>"*,(s0 -S<)

w0.HenceEq.207gives

3

spectivelytheconjugatesofthese.Hence

vesfor thetimefunction

A3e'"]

htforwardmannerto

uble-tunedcircuitofeitherFig.23 or24fora suddenly

dfrequency.


/ h t t p : / / w w w . h




ts havingthetransferfunctionsgivenbyEqs.213,

ssioncoincideswithEq.174for

owpasssecond-

ereforethe

envelopeof

appearance

nseof aband-

pplicationofa

dfrequencyis

elopethatisthe

thelowpass

e isinterested

chtheoscillatory

mputationsmay

oughcomput-

eofthelowpass

ehaveseen,is

atterninthe

ricallyareplica

bandpassnet-

w0.

nceofnet-

sa transfer

atternillus-

of Fig.31.Thepolesareuniformlyspaced


/ h t t p : / / w w w . h





s. Tosimplifythenumericalwork,the

saswellas theintervalbetweentheirlocusand

ltounity.

hasatransferimpedancewiththree

h fivepoles,andtheimpedanceofthethird

Specificallytherespectivetransferimped-

s+5)

s+5)(s2+2s+10)

esponse,thecoefficientsAkareparticularly

pe offunctionsinceallthefrequencyfactors

splanethatare anintegernumberofunits

straightuporstraight down(i.e.,inthe

thethree-polefunction213wehave

16)

e impulseresponse

-(e"/2-e^"2)^~'(217)

14,inspectionoftheappropriates-plane


/ h t t p : / / w w w . h




ponsebecomes

4e~'''+ e-*2')*-'=— W112-e^Ye~'

lefunction215,inspectionofthes-plane

ctlyto

20 +15e~"-6e~i2t+ e-^e-1

uivalentof

onsinthissequencefollowthesamepattern,

hatresultsfrom thistypeofpoledistribution

this chapter,thetransientresponseof

oleandzerodistributionsarethusseen tobe

allyifthepole-zeroconstellationsinthesplane

ometricsymmetryoruniformity.When

ithprescribedtransientbehavior,it iswel1

racteristictypesofresponsethatresultfrom


/ h t t p : / / w w w . h





urations,

ented.

hatthe

erFunctions

maydefinenumerousdriving-pointand

ngtoavarietyofpossiblechoicesthatcan

ationsofthe excitationandtheresponseand

wofunctionsis avoltageoracurrent.A

ervefurtherto emphasizethecharacteristics

er functionshaveincommon,andsimul-

herelationsgivenby Eqs.230arerelevant.

ffectivemethodfortheirderivationapplicable

wninFig.32in whichallelementvalues

y.Thevoltagesof therespectivenodesrela-

otedbyEuE2,E3,E4. Currentsintheseries

treferencedirectionare/12,I23,^34.A

enthedatumandnode1,feedscurrentIi

tthe right-handend,wehaveintheusual

+s4)

+s* +s5)

+5s4+s5+s6)

+5s4+6s5+1


/ h t t p : / / w w w . h




NSFERFUNCTIONS463

anreadilyforma considerablenumberof

toverinput).Tobeginwith,we havethe

s4+6s5+s6+s7

s4+6s5+s6+s7

s4+6s5+s6+s7

4+s5+s6

s4+6s5+s6+s7

es,becausethese arethenaturalfrequencies

circuitconditions.

emay alsoformthedimensionlessresponse

/E2,E2/Ei (232)

ywrite downforhimself,andobserveinci-

ios,andnot perchancetheirreciprocals,

tputoverinput).Ofthese,the first,fourth,

es,theybeing thenaturalfrequenciesofthe

1grounded,orthe zerosoftheinputimped-

1.Thesecondandfifthfunctionshave poles

enciesofthe circuitwithnode2 grounded,

unctionarethenaturalfrequencieswith

ltransferimpedancesarealsoobtained

,E3/I23,Ei/I3i(233)

gmentsofthenetworkofFig.32 ratherthan

firstthree functionspertaintothatportion

fterdeletionofthe branchesconfluentat

nctionsthebranchesconfluentatnode2 are

elastonefinallyinvolvesonly theparallel

eatnode4.

samenetworkananalogoussetofresponse

sourceconnectedbetweenthedatumand


/ h t t p : / / w w w . h




NSFERFUNCTIONS465

nttocurrentfedinto node2orinto node3

.SinceI\2is thecurrententeringnode2

tdivergenttowardtheleft, itisclear that

me nodepotentialE2mustequalthenet

bya sourcewhichisconnectedbetweenthis

s tosay,

234 wehave

+ssI2is

s4'E2~ l+s2

+5s4+6s5+s6+s7

+4s4+s5+s6

rfunctionsanalogousto theonesgivenby

4s4+s5+s6

5s4+6s5+s6+s7

2)

0s3+5s4+6s5+s6+s7

0s3+5s4+6s5+s6+s7

3+s4

6s2+ 10s3+5s4+ 6s5+s6+ s7

theratiosE3/E2and Ei/E2neededinthe

dthird oftheseimpedancesmustbetaken

ratioEi/E2 usedinthefourthone isthat

34,as maybeseenfromthefact thatall

ctions.

ntransferimpedancespertinenttoasource

.Again,usingrelations230and 234,wehave

2=2s+s3

3s2+s4


/ h t t p : / / w w w . h





+ 10s3+ 5s4+6s5+ s6+ «7

s2+ s4)

transferimpedances:

+ s4)

5s4+6s5+s6+ s7

s3+ 5s4+ 6s5+ s9+s7

+ s2)

s3+ 5s4+ 6s5+ s6+s7

3+ 5s4+ 6s5+s6+ s7

thatall ratiosmustbetransferfunctions.The

ctions241and 244are,ofcourse,thesame

enciesasbefore.

currentsourceisbridgedacross nodes2and

eadrivingpoint throughsolderingleads

wedenotethecurrentof thissourcebyI23

ncearrowpointstowardnode 3,thenin

-to-datumsourceswehave

oltageatnode4, forexample,becomes

and244yields

2)

s4+ 6s5+ s° + s7

s4+ 6s5+ s6+ s7


/ h t t p : / / w w w . h




NSFERFUNCTIONS467

s4+6s5+s6+s7

s5+s6)

s4+6s5+s6+s7

3 +s4)

0s3+5s4+6s5+s6+s7

betweennodes2 and4,andthedesired

ode 3,wefindthepertinenttransferim-

10s3+5s4 +6s5+s6 +s7

oltageE23inserieswith thebranchcon-

anyoftheexpressions247 or248wecan

rmationconvertingthecurrentsource /23

inquestionintoanequivalentvoltagesource

Sinceinthepresent examplethisconversion

from247forinstance

s3+5s4+6s5+s6+s7

roughthe 1-ohmresistanceattheright-

numericallyequaltoJS4,wemayalternately

short-circuittransferadmittanceforthis

ve(aspointedoutinthe closingparagraphof

sarethesameas thoseofanyofthe open-

sferimpedancefunctions.

riving-pointimpedancebetweennodes2

xpression249wemayderivetheratios

thedesiredresult isfoundtobe

s3+9s4+s5+2s6

s2+10s3+5s4+6s5+s6+s7(251)


/ h t t p : / / w w w . h




ifiedthatat t=0 thereisachargeof 1/2

ceatnode2anda currentof3amperesin

es2and3. Theinitialchargeof1/2coulomb

tteninto thecapacitanceatnode2througha

alue1/2being appliedatthisnode;andthe

we canimaginetobetheresult ofapplying

crossthenodepair2-3 oravoltageimpulse

einductancelinkingthisnodepair. (In

hasthevalueL23,then thevoltageimpulse

inedthroughaddingtogethera unitim-

hetransferfunctionEJI\,the1/2-multiplied

olvingthetransferfunctionEi/I2,andthe

onseforthetransfer functionEi/I23(orthe

esponseforthetransferfunctionEJE2^).

ctionsinvolvethesamepoles,the three

oninvolvethe samedampedsinusoids,albeit

andtimephases.Asmighthavebeenex-

s,theeffectofthearbitraryinitial conditions

ertheamplitudesandphasesof thedamped

ressionforthe networkresponse;theirfre-

sarethesameastheywouldbe forinitialrest

itionsaretobe consideredinthedeter-

nse,wethusseethatit isnecessarytodeter-

telychosentransferfunctions.Thediscussion

wsthat allthepertinenttransferfunctions

from thesamesetofbasicrelations ina

myandcircumspectiontothetotalcomputa-

ectofthings,a solutiontotheproblem

onditionsisfoundin thesamewayasis

conditions.

0/V3)<+10 cos10\/3<.

ceZ(s)attheinput terminals,andexpressitinthe form

actors.Makeans-planesketchindicatingthepole loca-

ros bycircles.Throughregardingthefrequencyfactors

etryismoregeneralthanthat ofanunbalancedladder,

themethodsdiscussedinthefollowingchaptertodeter-

g-pointortransferfunctions.


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_ u s e # p d - g

o o g l e




hembyinspection,computethe valueoftheimpedance

3,theradian frequenciesinvolvedinei(<).

(<)in thesteadystate.

ratioEt/E\as afunctionof«.Representits zerosand

s inpart(a),and computethecomplexvaluesofthis

Zand

intheexpres-

and«c(Z)

citanceele-

wthattheir

part(b).

he in-

bythecir-

verage

ower

hesecondtermin «i(<)iszero,(ii) thefirsttermin ei(i)

f thesevalueswiththenetpowerfoundin part(e).

theaveragevaluesof theelectricandmagneticstored

onsofpart (f).

edingproblem,supposeei(<)=10 sin10i.Compute

omputethepeakandtheaveragevaluesof theelectricand

Whatisthepowerfactor andtheaveragepowerabsorbed?

e circuitdrawnbelowhasthezero siandpoless2, ss

gs-planesketch.IfZ(0)— 1,whataretheparameter

100«,whatis e2(<)?Whatisthewidthw oftheresonance

werpoints?Whatisthe valueofQ?Checkthis value

thestoredand lossenergyfunctions.

ofaparallelLC shuntbranchloadedbythepurely

tunedcircuitandits complement.Thelowernetwork

coupled identicallytunedandequallyloadedresonant


G o o g l e - d

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_ u s e # p d - g

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rimpedanceE2/I1ineachcaseis tohavethepolepattern

-planesketchsuchthatcriticalcouplingyieldinga maxi-

teristicresults.

tthedata:(a) Midbandfrequency=uo=2t106,(b)

powerpoints=w=2,rl04,(c)impedancelevel=R =

hetransferimpedanceforthecapacitivelycoupledtank

ugatepairisthat determinedbyeitherofthetwoiden-

uitsconsideredseparately,whiletheotherpairmustbe

edbythe driving-pointimpedanceforthenodepaira-b,

etankcircuitimpedanceparalleledbythe coupling

teristhusseensimplyto lowerthenaturalfrequencyof

esecondpairof naturalfrequencies(poles)isalsoobvious

C2(<).Ifii(i)is aunitstep,howcanyou getthecorre-

utionjust found?


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_ u s e # p d - g

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circuitshownhere.Fromthe differentialequation

mofthisnetworkdeducethat

onsifE— IZ(s)whereZ(»),theimpedanceorfrequency-

thiscircuit,isgivenby

— u<,(<)>aunitimpulse,thenfort >0

eofacircuit isitscharacterizationinthetimedomain,

ction2maybe interpretedashavingafrequency-domain

nthefollowingexamples.

rcuitshownhere.Fromthe appropriatedifferential

uilibriumdeducethat

onsif/= EY(s)whereY(s),theadmittanceorfrequency-

thiscircuit,isgivenby

> 0isgivenby

te


/ h t t p : / / w w w . h




written

2

showthattheresults inthepreviousproblemapplyto

eoftheseriesRL circuit,andconcludethatthe frequency

ansform)ofastepofvalueE reads

essofyourconclusions.

esRCcircuit withtheadmittancefunction

-1/RC

tiontobea unitstepwiththetransformE(s) =1/s.

te

*)/(»)],_,»

eviousproblemshow thatthetimefunctioni(t)/C

nclosesunitareaand henceapproachesaunitimpulse

smaller.Thusshow thatthetransformoftheunit impulse

ofProbs.7 and10,obtaintheresponseof thecircuit

hroughuseof therelations

*)/(S)].-,*

tionalwayofsolvingthis problem.

olutioninProb. 7forthespecific valuessi=— 1,

hatis thevalueoftheresistanceR? Plotcarefully

for—3<< <3.Nowrepeatforthe valuessi=—2,

nclusioncan youdrawfromtheseresults?

Prob.7 forthespecificvaluessi =— 1,S2=/, with

orErea l)

rtheseassumptionsandcheckwiththe conventional

-calleda-cresponseof thissimplecircuit.


/ h t t p : / / w w w . h




DY.STATEANDTRANSIENT

eviousproblem,obviatethe necessityoftakingthe

essionsthroughuseof thefamiliarrelation

; s2= ja, S3=— ja

ob.7 andtherelation

* withAk=[(s -sk)I(s)],.,k

gerneeded,checkthesolutionfoundin Prob.13for&..= !.

cidentity

t= ja , S3= — ja

equency-domainrepresentation)ofthistimefunctionis

series RLcircuitof Prob.7forthisexcitationand u=1,

4.

n

s- sk)FW],.,l

of /(<),showthatdifferentiation(resp.integration)inthe

tomultiplicationofF(s)by s(resp.Checkwiththe

t,anduo(t),ui(0 asfoundabove(Probs.14,15,and 8,10).

llowingseries LCcircuithavingtheadmittance

ao1=1/LC,considerc(t) =Ee'1'for< >0,sothat E(s)

andao— 1,obtaintheresponseof thissystem,and

fProbs.13 and14.Whatimportantconclusioncanyou

elevanttoProb.12.


/ h t t p : / / w w w . h




7considerSi= 0,andcheckwiththestep-function

uitas foundbyconventionalmeans.

asan admittanceY(s)withthepole(crosses)andzero

wnintheaccompanyings-planesketch.Assumea/ud=

voltageexcitation.Inevaluatingtheresultingcurrent

ors(s,.— s,)asvectorsinthe splane,andobtaintheir

ckyourresultwiththatfoundby conventionalmeans.

representsaniron-coretransformerexcitedbythe

onlinearrelationbetweenprimaryfluxlinkagesand


/ h t t p : / / w w w . h





dasshownin thesketchontheright.Thus,belowsatura-

ctanceoftheprimarywindingisL =15/6thenry,while

0thof thisvalue.Thesaturationpointis characterized

ms

su =2,rX60.

mvalueofthenormalexcitingcurrentin theprimary,

rimarywindingresistance,computethemaximumralue

rrentfollowingswitchclosureatf =0.(c)Whatis the

witchclosesat<= t/o?

intheabovecircuitis

Show thattheresultingcurrentisgiven bytheexpression

S*)/(S)U«

S— S0

s

pplies with

i)

sinvolvedinthelastpart ofthepreviousproblemare

rclesinthes-plane sketchshownontheleft.

uesofthe parametersRandC,anddeterminei(0

<Sat t=0with


/ h t t p : / / w w w . h




tors(st— s,)byinspectionofappropriateS-planesketches,

fficientsAicwithaminimumofwastefuleffort.

emtheexcitationischangedto

)

chgiventherechanges.Specifically,evaluatet(0for

nextassumethata =1anda =1/2.Doesthesame

hangesinthe circuitparametervaluesareappropriate?

nofperfectresonanceresultingfora= a,andevaluate

tresponsewitha=1.

osituations:

ossibletohaveii(0■ anddeterminetheappropriate

eeither iior*2.

s excitedbyaunitcurrentimpulseat itsdrivingpoint,

s samepointisgivenbythe expression

fort>0

onZ(s),andsketchanetwork,givingelementvalues,

thefollowingsketchfind

sin s,andsketchthecorrespondingpole-zeroconfigura-

minetheanalyticexpressionfortheinstantaneousoutput

urrentii(flisa unitstep.


/ h t t p : / / w w w . h





etransfer impedance

<)is atriangularpulseasshownin theaccompanying

posedtoyieldan outputvoltagee^t)thatisa reasonably

hape.In ordertoinvestigatehowwellthenetworkmeets

neananalyticexpressionfortheerror

noftime.Expresstheapproximatemaximumvalueof

eoftheinputpulseamplitude.Hint.Inevaluatingthe

entiatetwicesoastoconvertitinto atrainofimpulses,

tioneitherthroughsubsequentintegrationorthroughdivi-


/ h t t p : / / w w w . h




eZu(s)ofacertainnetworkhasthe pole-zeroconfigura-

nyings-planesketch.Iftheinputcurrentis aunitim-

etwo ej(<)sketchesrepresentsapossibleoutput,andprove

oice.Statehowmanysuccessiveinitialderivativesofe2(t)

geisappliedtothe terminalsofalinearpassivenet-

eis

>0

ntadmittanceattheterminalsofthenetwork?(b)

ngthisadmittance.

figure,determinethetransferimpedance

<)cos t,findthetransformi?2(s)of theoutput

s-planesketchofits pole-zeropattern.Throughinter-

factorsasvectorsinthis plane,evaluatetheconstantsj4*

nspection,andformulatethissolution.Now,inter-

tternsofthesourceandthe circuit,determineacircuit

entvalues)andasource(definingit inthetimedomain)

enticalwith e*(t).

ereshowthat

a4s*

..• a4intermsof thecircuitinductancesandcapacitances.

tterintermsofthe coefficientsai...a4.


/ h t t p : / / w w w . h





hecircuitof Prob.31aredistributedasshownin tb*

a) Computethe

etervalues,(b)De-

ction| Z^ju)

w<3.

31withthe

Prob.32,deter-

0is(a)a unitim-

bs.31and32

eandtransient

angedbutthe de-

10,000ohms,what

ofLi, Lt,Ci,d?

kthetransientre-

imesfasterbutre-

whatdothe par-

Whataccompanying

steady-stateresponsefunctionZuXj'u)?

kofProb.31appropriateto thepoledistributionshown

he unitimpulseresponse,

.

-s*k)F*(s)],.,.k=Ak

/*(<)= /(0cosutf,has thetransform

as thetransferimpedance

ngthetransformI(s)= 1/s,isappliedatthe input,make

,showingthe polesoftheoutputvoltagetransform

ndingtimefunctione(t).Thecorrespondingbandpass

nce

2(s +]c*>)}


/ h t t p : / / w w w . h




on

akeans-planesketchof theoutputvoltagetransform

ngitspoledistribution.

fthe bandpassfilterislarge,thatis, ifuo» 1,show

m

juo,observethat

uo)]

outpute*(t) ofthebandpassfilterresultingfromthe

tendown,usingthe resultofProb.36(or theproperty

nctionresponseof thelowpassfilter.Checkthisresult

aluationofthetimefunctioncorrespondingtoE*(s),tak-

ificationspermittedbytheassumptionuo» 1.

assfilterofthe precedingproblemforacondition

itimpulseresponse,(b)the unitstepresponse,andnote

hesame,the principaldifferencebeingtheconstantmulti-

so.

rstpartof Prob.36with,a=As toshowthat

*(<)=—tf(t).That istosay,differentiationinthe s

timedomainto multiplicationby— i.Throughrepeated

thefunctionl/(s+1), findtheimpulseresponseofa

rob.31 withthetransferimpedance

metersforthisresponsefunction.

ciprocitytheoremortheprincipleofduality (whichever

t involvedinProbs.31,32,33,showthat thefollowing

oltageratioE»/Eiwhichis thesamefunctionasZ12M,

tparametervalues.

edingexercisesthat,if F(s)hasonlylefthalf-plane

alues,then/(<)—>0as t—>°°.On theotherhand,if


/ h t t p : / / w w w . h





chthat

nishfor<—><o

ctionresponseusuallyhassuchanF(s),and theconstant

tepartoftheresponse.Thefunction/*(S)— (Ao/S)evi-

<) — Ao,and,sincethistime functionvanishesforlargeI.

Ao/s)hasnopoleat S=0.Moreover,if Aoturnsoutto

ed step,thenwemaysaythatthe responseshowsno

ghastraightforwardextensionofthesethoughts,show

ansformwhichassuresazerosteady-stateerrorwhenthe

forn— 1,2,3,• ••is expressedbystatingthat

tramp; forn= 3itisaunitparabola , etc.

ywheretheresponseisamechanicaldisplacement,the

servohavingazero positionerror;n=2 yieldsonewith

,onewitha zeroaccelerationerror;etc.Theresponse

ystemwhichis »nF(s),whenitsatisfiestheabovecondi-

geofcomplexfunctiontheory)tohavea saddlepointat

ceof anetworkiswrittenin thepolynomialform

S*

ultsof theprecedingproblem,thatthesteady-stateerror

n(0 willbezeroif

hatZu(s)have asaddlepointoforder n— 1ata= 0.)

lexampleconsidertheimpedance

ationequalto theunitrampfunctiont'i(<)= «•(()•

utvoltageej(0,and observethatitsasymptoteactually

ntrastconsider

onzeroerrorremains.


/ h t t p : / / w w w . h




shorthandthatenablesonetowrite the

ngsystemsofsimultaneousequationsina

ncipalvalueliesinthe circumspectionthat

ness,andinthefacilitywith whichoneisthus

sualizethe significanceofmoreelaborate

braicoperations.Innumericalproblemsits

systematizationthatitinjectsintothecom-

ortcuts.It, therefore,isprimarilyatool

manipulations,butassuchitsusefulness

mountoftime andattentionrequiredonthe

erto understandthebasicprinciplesand

ntedout,theso-calledmatrixcorrespond-

ations

i

2nxn= y2

=yn

arrayofcoefficientsa,* inthesameorder

umnpositionsasthey appearintherelated


G o o g l e - d

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CIRCUITEQUATIONS

enequations.Unlikethedeterminant

ofits elementsa,kandhasdefinitenumerical

heseelements,thematrix [A]hasno"value"

uethati sprovidedbyitsoutwardappear-

onto theassociatedequationstowhichit

ationsequalsthenumberof unknowns

,xn),inwhichcasethematrixhas asmany

hatistosay,theassociatedmatrixis com-

d thenumberofitsrows orcolumnsisreferred

x.Thiscircumstanceisnotnecessarily

example,inCh.1wherewediscusstherela-

entsandloopcurrentsorbetweenbranch

tages,weencountersetsofequationswith

srespectamatrixagaindiffers fromadeter-

stalwaysinvolveasquare arrayofcoeffi-

onsquarematrixisone withonlyasingle

huswemaywritethesetsof quantities

,appearinginEqs.1,as matrices

olumnmatrices.

4,theshorthandknownasmatrixalgebra

t^ofEqs. 1inthe abbreviatedform

expressionmayberegardedasthe equivalent

ppropriateinterpretationforeachoftwo

atis meantbytheequalityofmatrices?

e productoftwomatrices,like[A].x]?


G o o g l e - d

i g i t i z e d

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_ u s e # p d - g

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questions,it followsfromwhathasbeen

wocanbeequalonlyif alloftheircorrespond-

ecessary(thoughnot asufficient)condition

hatbothmatriceshavethe samenumberof

rofcolumns.Since in5the right-handmatrix

ustturnoutthatthe productbea

n.Morespecifically,if 1and5 aretobe

esclearthatwemust have

)J

sinparenthesesintheintermediatematrix

tthis isacolumnmatrixlike x]ory]in 4

tfirstglancedoes notsuggestthisfact).

matrixwithcorrespondingonesin y]evi-

cationmadeevidentin Eq.6maybe

nemultipliesthe elementsintherowsof [A]

thecolumnofx]and addstheresults;and

lementsin theresultant(column)matrix

tc.rowsof [A],

ms,onemaysay that,informingtheprod-

d[B],resultinginthe matrix[C],onemulti-

ecolumnsof[B] inamannerthatis most

efollowingexample,

eelementsoftheproductmatrix

i3^31+014641

a13&32+014642

013633+aub43

023631+024^41

+023632+024642

023633+024643


G o o g l e - d

i g i t i z e d

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_ u s e # p d - g

o o g l e



CIRCUITEQUATIONS

mentin theproductmatrix(theelement

additionof productsoftherespectiveele-

andthe /cthcolumnof[B],ageneralformula

hesummationindex.Theequivalenceof5

e offormation,isreadilyrecognized,where-

ematrixEq.5as beingacompactwayof

ollowswithoutdifficulty.

ion,bytheway,onedoesnothaveto stick

ofthefirst maybemultipliedbycolumnsof

theproductA. B.Onemayequallywell

ctdeterminantthroughmultiplyingthe

ofB,orrowsby rows,orcolumnsbycolumns.

hemesmaythusbe usedindeterminant

mustbeconsistentthroughouttheevaluation

eedomresultsfrom thefactthatitis only

terminantthatmatters,and thisvalueturns

choneofthe fourschemesalthoughthe

mentsintheproductdeterminantare notthe

tionwheretheresultis amatrix,onlyone

y,sincetheelementsof thismatrixarethe

bythe exampleofEq.7is thefactthat,in

umberof columnsin[^1]mustequalthe

derthatthenumber ofelementsinanyrow

erofelementsinanycolumnof[B], aneces-

usfromthe wayinwhichtherow-by-column

ven matricesinaproductfulfillthis condi-

id tobeconformable.

e exampleinEq.7that thenumberofrows

olumnsin[B]maybe anything;butthatthe

ductmatrix[C]equalsthenumberofrows in

lumnsin[C]equalsthe numberofcolumnsin

nationofa matrix[A]withprowsand

q],anda similarnotationforthematrices

arksare summarizedintheequation

uctisplacedinevidencethroughthe

beingthesamein thetwomatricesforming


G o o g l e - d

i g i t i z e d

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_ u s e # p d - g

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A

ainingindexespand rcorrelatetherowsof

r]andthecolumnsof[6„r]with thecolumnsof

ormablemultipleproductis,forexample,

zedatoncefromthefactthatall adjacent

umberof rowsandcolumnsintheresultant

mthenumberof rowsinthefirstmatrix and

he last.

oftherule forformingamatrixproduct,

tativelawdoesnotapply;thatis:

awdoeshold,whichmeansthat,inthemulti-

oupthe termsinanywaywewish solongas

order.Thuswemaybegin themultiplication

]andworktowardtheleftin successivesteps,

tandworktowardthe right.Again,wemay

eproducts[apg]* [b9r]and[cr,]*[d,t],and.

thefirstoftheseby theresultofthe second.

matrixisthe sameinallcases,thecomputa-

herein liesoneofthe finerpointsofthis

rsue furtheratthistime).

metricalifitselementsfulfill thecondition

anyelementand itsmirrorimageaboutthe

efttolowerright) beingidentical.

matrix[A]is written[A]~1andisdefined

matrix,hastheindicatedstructureinwhich

paldiagonalareunity andallothersare zero.

a unitmatrixevidentlyreadxi= yi,

iplicationofany givenmatrixbytheunit

atrixunchanged.Hence,ifwemultiplyon

1,wehave

x]=[A]-1* y](15)


G o o g l e - d

i g i t i z e d

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_ u s e # p d - g

o o g l e



CIRCUITEQUATIONS

spondingtothis resultinthemannerthat 1

orm

=xn

ecognizedtobethematrixofthe inverse

e equationsthatrepresentthesolutionto

yof beinginverseisevidentlyamutualone.

egardset 1asbeingthe inverseof16,and

theinverseof[B];thatis:

nverseof agivenmatrixisthe problem

neousequationslike1or16.Using deter-

,onecancompactlyexpresstheelementsof

of thoseofagivenmatrix.Forexample,if

notedby A,asinEq. 3,anditscofactors

(seeArt.2,Ch.3)that theelementsofthe

aregivenby

nantof[B]isB, withcofactorsB,k,then

forthe elementsoftheproductmatrixyields

the unitmatrix14.

isclearthattheinverseof amatrixexists

eterminantisnonzero.Incidentally,there


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i g i t i z e d

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_ u s e # p d - g

o o g l e



A

terminantinthe firstplace,whichimplies

squarearray.Anonsquarematrixpossesses

oespossess aninverseissaidto benon-

s calledasingularmatrix.

e so-calleddiagonalform

xexceptthatthediagonalelementsarenot

ociatedequationsreaddnxi =yi,d22x2=2/2,

nbe invertedbyinspection.Onerecognizes

mply

ofadiagonalmatrixis againadiagonal

sdiagonalthat arerespectivelythereciprocals

the givenmatrix.

ghwritingits rowsascolumns,orvice

on,andtheresultis referredtoasthe trans-

nspositionofthematrix[A], Eq.2,yields

atrices4yieldsrowmatrices

matrixis indicatedbythesubscriptt,and

cesaredistinguishedthroughwritingx] and


G o o g l e - d

i g i t i z e d

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_ u s e # p d - g

o o g l e



CIRCUITEQUATIONS

enumberof rowsandcolumns,itmaybe

ntosmallersections,called submatrica.

maybewritten

umnsofthematrix[A] arepartitionedinto

dr andscolumns.Thatportionof [A]con-

thefirstprows andfirstrcolumns(the upper

trix[Opr],thatportioninvolvingelementsm

ts columns(theupperrightportion)is the

th.

ewisepartitionedasshownby

maybeevaluatedasthoughthesubmatrices

hatistosay,Eqs. 26and27yieldthe product

M»1M](28)

^]x [6r<]+[a,.]* [6JJ

ten

fthe columnsof[A]andof therowsof

nds)must correspondinorderthatthe

ppearingin 28allbeconformable.Thepar-

andofthe columnsin[B]is arbitrary.From

hesubmatricesin [A]andin[B] onecantell

umnsin thesubmatricesoftheproduct

slendscircumspectionwheredetailed

atematricesmustbecarriedout.

enumberofelementsina ratherextensive

at, afterpartitioning,oneencounterssub-

yofzeros.A matrixwhoseelementsareall

x.Apartitionedmatrixhavingsuchnull

form


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



RICESANDVOLT-AMPERERELATIONS491

nullmatrices asthoughtheyweresimple

osin thetoprowofmatrix30 arenull

ndrespectivelys andtcolumns.Similar

herzerosappearingin thismatrix.The

ischosentobe asquarearrayconsistingof

umberofcolumns.

sin30for themomentasonewouldordi-

ecognizethismatrix[A] tobeinthe diagonal

readilybeseen)thatits inverse,likethat

mply givenby

1inEq.23,althoughthe diagonalmembers

hodofmatrixinversionbecausetheyaresub-

elements.However,itisusefuli nanalytic

methodofindicatingtheinverseofmatrix30.

icesandVolt-AmpereRelations

the problemofsettingupthedifferential

equilibriumofalinearpassivenetwork,remov-

sothatwewill arriveataformulationthat

rtaintheoreticalconsiderationstobe taken

trictedpointofviewis essential,asmayalso

ituations.*

eneralprocedureisgiveninCh.2, itis

ywithreferencetoresistancenetworks.In

tion,it isnecessarythatweshowin detail

onspertainingtothebranches(expressed

7,Ch.2)maybeevaluatedfor inductances

sfor resistances.Forthesakeofconvenience,

tedbelow

)

nArt. 6ofCh.8 doesprovidethemeansfordealing

thereis lackingasyetacompactform expressingthetotal

olvedinobtainingequilibriumequationsforthegeneralcase.


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



UITEQUATIONS

theyrefer isreproducedhereasFig. 1.

alformthatapassivebranchwith associated

smaytake.Sincejk andvkarethenet cur-

quantities(j* +i,*)and(t>*.+e.*)are

siveelement(resistance,inductance,orcapaci-

hecurrentsandvoltagesinthe passive

dbyEqs.32 and33.Thefirstof thesesym-

tage

ementof

onof

s;the

r does

put

ex-

gle

reas-

nsec-

let ussaythatX areinductive,parere-

X +p+a beingequaltob.The number-

ver,iscarried outinsucha fashionthat

uctances,numbersX+ 1toX +preferto re-

+p+ lto\ +p+a =breferto elastances.

hmaybe mutuallycoupledwithevery

,thematrixofself-andmutual-inductance

hediscussioninArt. 4ofCh.8,called the

,hastheform

ociated

s.

cebranches,ontheother hand,cannotbe

rametermatricesmusthavethediagonal

sistancematrixisgivenby


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



TRICESANDVOLT-AMPERERELATIONS493

matrixiswritten

s,anelementr,-ors,-is simplytheresistancein

rafsof thesingleelement(passivebranch)

scriptrefers.

assiveelementsintermsofthecurrentsin

ssedfortheinductancesby

k),

), i-X+1, ••• ,A+p(38)

he equations

), i=A+p+1, ••b(39)

erationsofdifferentiationandintegration

ofmutualcouplingbetweeninductive

oltagedrop(p,-+e„)in oneofthese(i.e.,for

n generaluponallofthe passivecurrents

ThatiswhyEqs.37 involveasummation

vebranches(referencetoEqs.30inArt. 4,

intheinterpretationof37).Eachof Eqs.

volvesonlyasingletermonthe right-hand

ropina resistanceorelastancedependsupon

lone.

9 maybecombinedintoasinglematrix


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



CIRCUITEQUATIONS

thecolumnmatrices

atrix

etermatrices34,35,and 36areembedded

[D]is writteninpartitionedform).The

becomeassociatedwitheachelementin

ncemultiplicationofamatrixby ascalar

thematrixby thatscalar(asmaybe seen

rixmustvanishifthescalar iszero,anda

nallelementsarezero).

essingthevoltagedropsin thepassive

rrents intheseelements(equivalenttoEqs.

re givenbythesinglematrixequation

dicatedmatrixoperationsonethus obtains

esymbolicEq. 32.(Theuninitiatedreader

[D] completelyandcarrythroughtheindi-

erto understandthisresultandappreciate

aysimilarlybe evaluated.Tothisend

tricesareconsideredintheirinverseformsas

matrix


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



TRICESANDVOLT-AMPERERELATIONS495

x

gonalmatrices,theirdiagonalelementsare

ediagonalelementsin[r] and[S].Asshown

eelementsin [7]arenotso simplyrelatedto

determinantof[I]beA,with cofactors

oneither7or Amaybeinterchangedsince

esymmetrical.

edinverserelationsare expressedby

k),

combinedinasingleequationthrough

ix(inverseto[D])

nsforthecurrentsinthe passiveelementsin

n theseelementsaregivenbythematrix

2)


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



CIRCUITEQUATIONS

andrepresentstheexplicitevaluationofthe

ibriumequationswecannowfollowpre-

tin Art.8ofCh.2 forresistancenetworks

s.44,45, 4Gfortheloop basisandbyEqs.

sis.Thecentralequationineachof these

thevolt-ampererelationsforthe branches,

dputtinginto matrixforms43and52.The

achgroupexpressesrespectivelythepertinent

chvariables(currentorvoltage)in terms

es.Theserelationsinvolvethetie-setand

llshownexthowthesemayconveniently

onsandcombinedthroughstraightforward

52 toobtainthedesiredresults.

efo„

«»*

thismatrixarethecoefficientsina Kin

sincetheypertaintotheselectionofa cuts

llnormallybeeither±1or zero,depend

nchdoesor doesnotbelongtothe cut

chrowhas belements;buttherearec

mberofindependentcutsetsor nodepj

he discussioninCh.1thatn =nt

umberof nodes.

t theelementsinanycolumnofthe mi

uationexpressingthepertinentbra

henode-pairvoltagesthatareconsistent

s.Therefore,if wewritecolumnmatrii


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



ONTHENODEBASIS

and voltagedrops,andexpresssimilarly

dvoltagesourcesas

es41can beseparatedasindicatedby

=M+[«j(56)

pairvoltagevariablesintheformofa column

ationsareexpressedinmatrixformby

52forthebranchescanbe written

1* [e,\(59)

termsof thenode-pairvoltagesaregivenby

uationsaretheKirchhoff-lawEqs.58

ode-pairvoltagesas variables.Oneobtains

tingtheexpressionfor [v]fromEq.60into

ation for[j]intoEq. 58.Afteraslightre-

onefinds

* ([<,]-[D]-1* [e.])=[in](61)

e,representingthenet equivalentcurrent


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



CIRCUITEQUATIONS

pairs,isabbreviatedas acolumnmatrix

estructureofthismatrixas expressedby

withintheparenthesisrepresentscurrent

ebranches,whiletheterminvolving

ransformationofvoltagesourcesassociated

1)i ntoequivalentcurrentsources,the

oppositereferencearrowsassociatedwith

sexpression,therefore,representsnetcur-

es,andis tobethoughtofas combiningthese

x.Multiplicationofthematrix [a]intothis

nacolumnmatrixwhoseelementsarealge-

ourcesaccordingtothe groupsofbranches

the branchesassociatedwiththepertinent

nthis resultantcolumnmatrix[in]arethus

e-paircurrentsources.

andsideofEq. 61isfacilitatedthroughan

thetriplematrixproduct[a]* x[a]t.

achievingthisendis thepartitioningofthe

to groupsofX,p,<r,thus:

stsofthefirst Xcolumnsin[a];[anp]repre-

pofpcolumns,and[anc]containsthe lasta

or[D]1 giveninEq.51,one thenfinds

n„j ana]*

«np]x [9]* KpL+x[c]«[anc]tP(64)


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



ONTHELOOPBASIS499

elyasthereciprocalinductance,theconductance,

metermatricespertainingtothenodebasis.

sourcematrix62,theequilibriumEqs.61

amiliarform

=[in](66)

present discussionaretheevaluationof

x[*n]asgiven inEq.61,andthe expressions

etermatrices.Thesearegivenin termsof

trices[7],[g],and[c]throughtriple-matrix

portions(andtheirtranspositions)ofthe

ecut-setschedule.Theformationofthese

ntheequilibriumEqs.66 isthusin every

ystematic,andstraightforwardprocedure.

sareasetof simultaneousdifferentialequa-

ntaneousvaluesofthevariables.

theLoopBasis

tirely analogoustothatjustdescribed.

duleischaracterizedbythematrix

yrow arethecoefficientsinaKirchhoff

etheycorrespondtotheselectionofatie set

closedloop).Their valueswillnormallybe

ngon whetherapertinentbranchdoesor

setdefinedbya givenrow.Eachrowhas

nly Irowssincethis isthenumberofinde-

tshouldberecalledfromthe discussion

— nt+I.

t theelementsinanycolumnofthe matrix

uationexpressingthepertinent branchcurrent

ntsthatareconsistentwiththetie setsdefined

hatcirculateuponthe closedpathsdefined

wemakeuse ofcolumnmatrices54and55,

qs.5G,and writetheloop-currentvariables


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



CIRCUITEQUATIONS

atrix

ationsareexpressedinmatrixformby

43forthebranchescanbe written

0)

terms oftheloopcurrentsaregiven by

uationsaretheKirchhoff-lawEqs.69

oop currentsasvariables.Oneobtainsthis

theexpressionfor[j] fromEq.71into

ation for[v]intoEq.69. Afteraslightre-

onefinds

c]-[D]*[*.]) =N

e,representingthenet equivalentvoltage

isabbreviatedasa columnmatrix

estructureofthismatrixas expressedby

withintheparenthesisrepresentsvoltage

ebranches,whiletheterminvolving[D]* [i,J


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



ONTHELOOPBASIS

tionofcurrentsourcesassociatedwithbranches

voltagesources,the minussignarisingfrom-

owsassociatedwithi,kande,*.The paren-

re,representsnetvoltagesourcesforthe

ughtofascombiningtheseinto asinglecolumn

ematrix[ftinto thiscolumnmatrixyields

seelementsarealgebraicsums ofthebranch

oupsof branchesformingtiesets(theseare

ithpertinentloops).Theelementsin this

;]arethusseen tobeequivalentloopvoltage

andsideofEq. 72isfacilitatedthrough

ofthetriple-matrixproduct[ft* [D]* [ft,.

achievingthisend isthepartitioningofthe

ntogroupsofX,p, a,thus:

softhe firstXcolumnsin [ft;[fii„]represents

columns,andcontainsthelasta columns.

eninEq.42,one thenfinds

ft,]*

[r]* [ftj,+ [0,,]* [«]* [0la]tp-1(75)

yastheinductance,theresistance,andthe

ricespertainingtotheloopbasis.In termsof

x73, theequilibriumEqs.72takethe some-

[ei]


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



CIRCUITEQUATIONS

present discussionaretheevaluationof

x[e;]asgiven inEq.72,andthe expressions

etermatrices.Thesearegivenin termsof

trices[I],[r],and[s]throughtriple-matrix

portions(andtheirtranspositions)ofthe

etie-setschedule.Theformationofthese

ntheequilibriumEqs.77 isthusin every

ystematic,andstraightforwardprocedure.

sareasetof simultaneousdifferentialequa-

ntaneousvaluesofthevariables.

atsymmetryofthe parametermatrices

omesaboutifthedefinitionsof thenode-pair

rechosento beconsistentwiththeKirchhoff-

thatiscommonlymetbutis bynomeans

esgiveninthetwoprecedingarticlesare,

general,fortheysatisfytheconditionsleading

method,forexample,Eq.58 expressing

Eq.GOdefiningthenode-pairvoltagevari-

eyinvolvethesamea matrix(cut-setsched-

method,theKirchhoffvoltage-lawEq.69

opcurrentsareconsistent,fortheyarebased

edule(/3matrix).

isthefact thatonemay,onthenodebasis,

rs forthecurrent-lawequationsandanalto-

edefinitionof thenode-pairvoltages;or,on

ooseoneset ofloopsforthevoltage-lawequa-

irculatorypathsfortheloopcurrents.Specifi-

involvetwodifferentamatricesorcut-set

d71may involvetwoentirelydifferent/J

es(solongas theschedulesusedpertainto

se).Inmostinstances,however,itis advan-

onsistencyconditionsandobtainsymmetrical

eforetherelationsasgiveninthepreceding

appropriateandcanreadilybegeneralized

oftheproceduregivenin thepreceding

erthecircuitofFig.2(a)whichinvolvessix

sonthesamemagneticcore.Anappropriate

wninpart(b) ofthesamefigure.It iseasily

hediscussioninArt.4,Ch.8) thatthescheme


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



ritiesofthecoils bymeansofdotsis appli-

ement,andthatthe systemofdotsinthe

sistentwiththe physicalarrangementindi-

einductivebranchesofthiscircuitdiagram

ipsofthesearrows areatthedot-marked

mpleexpedient,allmutualinductancesbe-

numericallypositive.Ifweassumethatall

mentofcoupledcoils(a)and theirpertinentcircuitcon-

ncesequaltounity,andthatall mutualinduct-

f(thereis nosenseinusingmore arbitrary

onehasthe branch-inductancematrix

ce,whichisbridgedacrossanodepair,

yvoltagesourcesinseries withbranches(as

moreinterestingin thisexampletoconsider

eneratebranch;thatis, oneforwhichthe


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



CIRCUITEQUATIONS

ntiszero.Forthebranch numberingindicated

insthegraphshowninFig.3.

mweareparticularlyinterestedinthecur-

eft asafunctionofthe sourcevoltage,itis

n suchawaythatbranches7 and8become

othe

tothegraph

. 4,forwhichbranches1,7, 8arelinks,is a

eidentificationoflinkcurrents withloop

ciates*'i withthesourceand12 withtheload.

tie-set scheduleisnowrecognizedby

givenbythefirstsix columns.Notingthe

ncematrixinthe Eqs.76,wenextevaluate


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



henfoundasshownbelow

ontheloopbasis arethusseentobe given

/ (83)

=d/dtis stillused.Sinceweare onlyinter-

odideatoeliminate13 immediately.Todo

o-calledaugmentedmatrix

ationsof itsrows(equivalenttomaking

Eqs.83)i nsuchawayas toproducezeros

dcolumnexceptthelast.

multipliedelementsofthethirdrowtothe

row; andthenaddthe(— l/2)-multiplied

therespectiveonesofthesecondrow, there


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



CIRCUITEQUATIONS

ognizethatthecircuitofFig. 5involving

hthevaluesindicated,hasthesameloop

emay,

plecir-

e one

termi-

con-

hecir-

hree

coupled,andfortheindicatedreferencearrows

ranch-inductancematrix

othat

e terminal

veconductancevaluesinmhosasindicatedin

acterizedonthenodebasis.

-


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



ductancematrixforthebranchnumberingin

oosea node-to-datumsetofvoltages,

atumanddenotingthepotentialsofnodes A

de2. Theappropriatecut-setschedulethen

wewillassumeto besteadysinusoidswith

olumnmatrix

o55is identicallyzerobecausethereareno

dwithanyofthebranches.

wemustformwhichbyEq. 51


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



CUITEQUATIONS

rowsandcolumnsenterintothe evalua-

erifyby inspection,notingincidentallythat

sint= — cost.Onethusobtainsforthe

tequivalentcurrentsourcesfeedingthenodes

)1

m(/ +45)J

onintoEqs.65nextyieldsforthe param-

vebeenwrittendownbyinspectionofFig.6.

tionsforthisnetworkbecome


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



S

es55pertainingto thebranchesasindi-

e,x]containthe firstXelementsinthe

,],thesucceedingp elementsarerepresented

d[e,„],and thelastaelementsare combined

,a\.

d[ei], Eqs.62and73,pertainingto node

riatelyresolvedintoadditivecomponentsas

(100)

ressionfor[in]givenin Eq.61permitsthe

* [e,x])(102)

e,p])(103)

[«„]) (104)

]givenin Eq.72maybedecomposedinto

05)

.J)(106)

])(107)

matrices[*n]and[e{] thataredistinguished

ottobe confusedwithsubmatrices)arethose

ourcematricesthat arecontributedbyactual

sassociatedwiththeinductive,the resistive,

espectively.Theseparateexpressionsfor

essunwieldythanthosefor [in]and[ei].

ourcesassociatedonlywithonekindof ele-

nce,orcapacitance)inwhichcasetwo ofthe

yzeroandneednotconfusethe calculations.

above,therearenosourcesassociatedwith

dsothree-fourthsofthe spaceoccupiedby

ceoccupiedby94could besaved.


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



CIRCUITEQUATIONS

asizingisconcernedwiththewayinwhich

oblemof Fig.2isdealt with.Whereverwe

ethatisnot inserieswitha passiveelement

notinparallelwitha passiveelement,we

evisingthe circuitaccordingtothediscussions

f consideringthesourceasadegenerate

xampleabove.Thelatterschemepreserves

network,afeaturethat maybeimportant

tationoftheanalysis.It iswell,therefore,

ityoftreatingsourcesinthesevariousways.*

heloop-parametermatrices[L],[R],[S]

dexest andfcmayassumeanyintegervalues

theequilibriumEqs.77 inthefollowing

m:

=en, i=1, 2, ••• , I(108)

ompactformfortheseequations,thereader

notationinvolvedthroughwritingthemout

etofpaper.Thus fori=1 heshouldwrite

andsumthatcorrespondsuccessivelyto

tethesetoen.He shouldthendothesame

rthdowntotheequationfori=I.Hew ill

andclarityhowthis systemofequations

ndhowthe notationinEq.108is tobeinter-

ghthedetailedevaluationof thesesame

eequivalentmatrixform77,writingout

yandcarryingout theindicatedmatrix

ons.Afacileunderstandingoftheequiva-

algebraicformsoftheseequationsandtheir

etationthusgained(andnotachievable

ethod!)isessential indevelopingone's

houtdifficultythefollowingdiscussion.

esen, e;2,.••,e« asbeingactualvoltage

of thenetworkandregardtheloopcurrents

thesesamelinks(whichisappropriate

adingtosymmetricalparametermatricesare

oProbs.23 and24ofCh.2.


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



eexpression

09)

taneouspowerdeliveredtothenetworkby

udythisflowofenergythroughoutthenet-

theexpression109using Eqs.108.Visual-

enoutas suggestedabove,wemayformthe

tiplyingtheequations(onboth sides)suc-

ndadding allofthem.

tionsignwe canindicatethissetofopera-

multiplyonbothsides ofEq.108byi,- and

from1toI. Ontheleftweobtaina double

alreadyinvolvesasumwithrespectto the

en

iif ikdt)=£ehii(110)

otethatthetimedifferentiationandintegra-

tionsas

seisusedto denotetheloopchargeorindefinite

urrentasindicatedin

equivalentto

(115)


/ h t t p : / / w w w . h




CUITEQUATIONS

rddifferentiationweseethat

rentiationareinterchangeableoperations,

vativeofaproduct

osumsweinterchangethe summationindexes

lesinceeachindependentlyassumesall

),andmakeuseofthe symmetrycondition

rthatthetwosumsare identical.Hence

fEqs.110 and115isestablished.

Ch.7 revealsthatourpresentEgs^lll,

nta generalizationofthepreviousresult

circuit, andthattheJunctions2F,T, and

instantaneouslateofenergydissipation,

ftheenergystoredin themagnetic^fields

ances,andtheinstantaneousvalueofthe

c fieldsassociatedwiththecapacitances.

econservationofenergythroughshowing

gysuppliedbythesourcesequalsthe sumof

onin thecircuitresistancesandthe time

toredenergy.

spowerwhileTand Vrepreseru^eaergj.,

ionsF, T,andVas theenergyfunctions

rk.Specifically,TandV arereferredtoas

whileFisalsocalledthe lossfunction(first

analysisbyLordRayleigh).Iu-wderto


/ h t t p : / / w w w . h




ormforthethreefunctions,afactor1/2 is

tioninthe"expressionforF aswellasin those

asoniti s2FnotF thatrepresentsthetotal

gydissipation.

omeaccustomedtothenotationusedin

lloccasionallytowriteoutanexpressionof

xample,

— •+Liiiiii

L2faii

pressionisa summationontheindexk

exiremainsconstantatthevalue1; thesecond

ononfcfrom1to Iwhile* =2,andso forth.

addthetermsbycolumns,wesee inci-

natelysumoni from1toI holdingkconstant

.In otherwords,thedoublesummationis

theindexesi andkindependentlyassume

, asstatedabove.

parealsowithEq.154of Ch.3andwith

thusseentobehomogeneousandquadratic

the loop-currentvariables.Asmentioned

issortis calledbymathematiciansaquad-

ctionsF,T,andV characterizingalinear

eentobe quadraticformsintermsofthe

etheenergyfunctionsintermsof thenode-

StartingfromEqs.66 pertainingtothenode

mentsinthe matrices[T],[G],[C]byr,*,

ebraicformfortheseequationsreads

=ini, i=1, 2, ••n(122)

enode-pairvoltagesandini• ••innare the

entsources.Sincebothsetsofquantities

s, thetotalinstantaneouspowersuppliedto

etof currentsourcesfeedingthepertinent

n


/ h t t p : / / w w w . h




CIRCUITEQUATIONS

hroughmultiplyingthesuccessiveequa-

elybyei,e2,en andadding.Theresultis

**e, fe. dt)=£

ctions'

susedto denotethenode-pairfluxlinkages

ofthe node-pairvoltagesasindicatedin

isanexpressionofthe conservationof<

129)

orthisconclusionbeingentirelysimilarto

sis.

etworkmaybethoughtof asexcitedby

thelinks,wecansubsequently-assumethe

ngthevaluesof theresultinglinkcurrents

ytherest ofthevoltagesandcurrentsthrough-

ofthe network,however,isregardedas

oop-currentvariablesorintermsof node-

ordingtowhetherthesourcesareconsidered

urrentsrespectively.Inasituationof this

2,113havevaluesthat areidenticalwith

25,126,and127. Theformerexpressthe

nergiesintermsof loopcurrentsandloop

thelatterexpressthese samevaluesinterms

luxlinkagesasvariables.


/ h t t p : / / w w w . h




e sourcesareanytimefunctions.Letus

steadysinusoidsas theyareinmanyprac-

oopbasiswethen write

at)(131)

conjugatevalue.Preparatorytomaking

ons111,112,113 forF,T,andV,we compute

-jai) (132)

k+Ijk)

33)

realpartof"asused inpreviousdiscussions.

31 that

134)

,112, 113thengivesforthe energyfunc-

dystate

./*](135)

/J(136)

)

ttheResign isnotneededineach firstterm.

)

al inspiteofits complexvariablesIiand

mentisreadilygiventhroughshowingthat

dsideofEq.138is self-conjugate;thatisto

e.Obviously,ifanumberequalsitscon-

bereal.Consideringtheconjugateof this


/ h t t p : / / w w w . h




CIRCUITEQUATIONS

s

weinterchangethesummationindexesiand

bolscouldbe usedintheirplace.If wethen

seenthat138and139 areidentical.

nsF,T,V,accordingto theEqs.135,136,

faconstanttermanda double-frequency

h.7forthesimple RLCcircuit.Theconstant

veragevalueoftheenergyfunction.Wethus

onsequivalenttoEqs.136and 137aregiven

sinsteadof loopcurrents.Theseareless

esultsbecausetherelation forTgiventhere

ssibility ofmutualinductivecoupling.Since

opcurrents,donottraverseanycommon

bilityofmutuali nductivecouplingisnotcon-

venin Ch.7,thesumsappearingthere(see

volvenocross-producttermsasdo theones

aretermsare present.Forthisreasonitis

econstanttermor averagevalueisgreater

eamplitudeofthe oscillatorycomponent,

nobviouslyrequiressincetheinstantaneous

dotherwisebecomenegativeduringsome

physicallyimpossible.

eralexpressions135,136,137considered

viousalgebraicallythattheconstantterms

amplitudesofthe pertinentoscillatorycom-

dentpurelyalgebraicproofcanreadilybe

efunctionbyitself,oneintroducesalinear

ableswhicheliminatesthecross-product

iredresultisagainobvious(as pointedoutin


/ h t t p : / / w w w . h




^ "517

t.7, Ch.7).Thealgebraicdetailsinvolved

however,notjustifiedatthis point.

gousresultsintermsofthe node-pair

sideringthefunctionsF,T,V,as givenby

e forthenode-pairvoltagesandfluxlinkages

ke~iut)(144)

iE^2*'](145)

*"](146)

qs.125,126,127yields

Go*,-ftl(147)

ECa*A| (148)

r,*^J(149)

neachfirst termforthereasongivenin

goussituationon theloopbasis.Thefirst

evaluesandthe secondtermsaredouble-

eamplitudescannot(forapassivenetwork)

uesoftheconstantterms.Theseaverage

mplexamplitudesofthenode-pairvoltagesare


/ h t t p : / / w w w . h




CIRCUITEQUATIONS

oobtaingeneralexpressionsforactive,

erinthesinusoidalsteadystate.Tothis end,

umEqs.108for theassumptions

53)

ponentialfactor,onehas

, •• , I(154)

ueuponbothsidesandrearrangingthe terms

155)

cordingtoitsfundamentaldefinitiongiven

inedthroughmultiplicationby7,./2and

t,thus:

* -^EW*)

0,141,142thisgives

+j2o>(V„-rav)(157)

tsobtainedin Ch.7forthesimple circuit

ectorpoweristheaveragepowerdissipated

maginarypartor theso-calledreactivepower

encebetweentheaverageenergystoredin

storedin themagneticfields.Whenthese

esareequal,thesourcesarenot calledupon

ngeofstoredenergy,andthenet reactive

Ch.7,thereactivepowerQavisa measure


/ h t t p : / / w w w . h




sourcesarecalledupontoparticipateinan

rgy.Theactualaveragepowerconsumedby

dactivepowerPav.

etalsoEq.115,expressingtheconservation

steadystate throughsubstitutingtheex-

rF, T,V.Ifweobservethat thelasttwo

n136 and137donotcontributeto these

upthesinusoidaltermsin asinglesum,this

ves

(r* +jwL a+^)/<J*1

E(f l«+jwL ik+J*}**](162)

thedoublesumintotwo singlesumsinorder

ethecurvedbrackets,accordingtoEq.154,

e

nstantaneouspowersuppliedbythe sources

ragepowerdissipatedbythe circuit)plusa

d.

edouble-frequencysinusoidwecansee,for

orkisa balancedpolyphasesystem,thisterm

cevoltages£?,•andthesourcecurrentsare

eequallyspacedin timephasesothatthe

erallsourcesvanishes.Inanybalanced

linstantaneouspowerisconstantandequal

umedbythenetwork.


/ h t t p : / / w w w . h




CIRCUITEQUATIONS

eresult163if thenetworkisexcitedbya

ne beEi,wehavein thisspecialcase

4)

ceanddenotingthe inputadmittanceangle

)(165)

ten

os(2w<+*>) (167)

theamplitudeofthedouble-frequencysinus-

fthe vectorpower.

andLagrangeEquations

owthatLagrange'sequations,which

asystemin termsofitsassociatedenergy

htheKirchhoff-lawequationssofar asthe

Weneedfirst somepreliminaryrelations

rom Eqs.Ill,112,113for thefunctions

currents.If wedifferentiatepartiallywith

current,wefind

silybeobtainedif oneconsiderstheperti-

ompletelyasT isinEq.121. Itisthen

opcurrent, sayi2,iscontainedin allterms

ondcolumn,andonlyintheseterms.Hence,


/ h t t p : / / w w w . h




OFFANDLAGRANGEEQUATIONS521

withrespectto i2,noothertermsare in-

+ If s*a+•••+

(171)

m withL22yieldsafactor 2becausethe

However,sinceL,* =L*,.,wecanrewrite

•+L2iit)(172)

.Equations168and 170areobtainedin

ee, thesummationinvolvedisasimple

otallywith respecttotime,wehave

tenas

tain

voltage-lawEqs.108nowshowsthatthese

en

(176)

tageequilibriumequationsareexpressed

ctions,isknownastheLagrangianequations.

arehereobtained,itis clearthattheyare

-lawequationsalthoughtheiroutward

ethisfact inevidence.


/ h t t p : / / w w w . h




CIRCUITEQUATIONS

smayalternatelybeexpressedintermsof

variables.Toobtainthisr esultwebegin

forF,V,Tand form

allywithrespecttotimeand rewritingEq.

rchhoff-lawEqs.122mayberewritten

ngianequationsexpressingthenetwork

associatedenergyfunctions.

thatEqs.176and183 aredualformsof

ustastheKirchhoffEqs.108 and122are

gincomparingthesetwoequationstonote

interchangeplaces,asshouldbeexpected

areduals.

unctions

esultsofthischapter,it isasimplematter

timpedanceofanetworkinterms ofits

ons.ThusEq.157forasingledriving point


/ h t t p : / / w w w . h




FUNCTIONS

yields

(184)

each sideofthisequation,

v)(185)

84 byExEx=|Ei |2orbothsides of

srespectively

eser esultsreads

,)]*.,(188)

7av)]/,_i(189)

188 oneshouldconsiderFav,V^y,T,

ofvoltagesasinEqs.150, 151,152,fortheir

readilyrecognizedtobe proportionaltothe

thedrivingpointsinceall othervoltages

portionaltoE\.I nEq.188wesay that

sbeingevaluatedper voltatthedriving

etationofEq.189we associatewithFav,

40,141,142in termsofcurrents.Sinceall

arlyproportionalto I\,theaverageenergy

roportionaltoI\2.Theirvaluesperampere

eregardedasnormalizedvalues.Equation

pointimpedanceintermsofthese normalized

gyfunctions.

h. 7wherethesesameexpressionsfor

ceandimpedancefunctionsarederivedin

cuit, onerecognizesbyinspectionthata

forwhich theaveragestoredenergiesare

nceoradmittancethen becomespurelyreal.

edmagneticenergypredominateswhenthe

nceis positive,whileanegativereactive

ed electricenergypredominates.Thusthese


/ h t t p : / / w w w . h




CIRCUITEQUATIONS

dancearemoreclearlyand directlyrelated

fthe network.

inasensepermitthe samesimplecorrela-

oradmittanceandthephysicalnetworkto

areotherwiseonlypossiblewiththe simple

t. Theyare,however,restrictedinthat

nceisexpressibleonlyfor pureimaginary

referredtoasreal frequenciessincetheycorre-

adyamplitudes).Inthe followingweshall

oughtheintroductionofa relatedsetof

esignificanceforanycomplexvaluesof the

ju;andsimultaneouslyweshallgeneralize

allpossibletransfer impedancesoradmit-

ng-pointfunctions.*

pequilibriumEqs. 108,letussubstitute

90)

nofthe exponentialfactor

2, ••• , I(191)

thatresultfori =1,t= 2,andsoforthwe

byI\,72,• ••,Iiand add.Ifweintroduce

94)

5)

msforT0, F0,VoshowninEqs.192, 193,

nterchangethelettersi andfc,whichare

rstintroducedbyO. Brunein1930asa preliminary

itivereal characterofadriving-pointimpedancefunction

sufficientcondition foritsphysicalreadability.


/ h t t p : / / w w w . h




FUNCTIONS

es,andthennotethesymmetrycondition

ortheparameters.AlthoughT0isthe only

sthathas thedimensionsofenergy(F0is

V0hasthedimensionsofthetime rateof

forthe sakeofsimplicityrefertoall threeas

ationtothefunctionsTav,Fav,Vavisdis-

ractfromEq. 195somerathergeneral

ng-pointandtransferimpedances.Inthis

ll thatinmostcasesweare notinterestedin

oopsofanetwork.Inorder,however,to

xiblestate,weshallassumethat,ofthe I

p willbeconsideredasbeingpointsof access,

egerfrom1 toI.Thesep pointsofaccessare

oltagesourceslocatedinpofthe loops.The

umeto beenclosedinaboxwith onlythe

broughtout.

yin thecurrentsatthesep terminalpairs,

er currentsinvolvedinEqs.191.In order

wthiseliminationisdone,wemayassume,

striction,thatthepointsofaccesscorrespond

bbreviation

hefirstploops havenoexcitation.

quationswenowrepresentin theparti-

equilibriumEqs.191 tobewritten

p

fp+U^;=0

0


/ h t t p : / / w w w . h




UITEQUATIONS

containstheelementsofthe firstprowsand

heelementsofthefirst prowsandthelast

p,andsoforth.The columnmatrices

ionedasindicatedbytheforms

mnsubmatricescontainingthefirstp

]containsthelastq elementsin[7],and[0]

eEqs.197maythenbe writtenmorecom-

onsmaybesolvedfor [/,]giving

st equationyields

=l^pl

redsetofequationsinvolvingonlythe currents

pairs.

mpedancematrixoforder p

=

intheequivalentalgebraicform


/ h t t p : / / w w w . h




FUNCTIONS

ftheEqs.191appropriateto thesituationin

presentinthefirst ploopsonly.Thecoeffi-

nsaredeterminedfromtheJ "<j(Eq.196)

amannerindicatedbythe matrixEq.204.

1to Eqs.205maybedescribedasa process

ngtheinaccessibleorunwantedcurrents.

re derivedspecificallyontheassumption

ndthe7'sare responses,theycorrectlyrelate

ts,eventhoughsomeorall oftheI'smay be

glysomeorallofthe E'sbecomeresponses.

5areregardedas sources,thentheresulting

veterminalpairsareexplicitlygivenbythese

cumstancestheterminalpairsareallopen-

onthez# arereferredtoas asetofopen-

ansferimpedancescharacterizingthepter-

parewiththeanalogousquantitiesdescribed

ncenetworks).

ynonzerocurrentsource,then

qual invaluetoonereferenceampere,

Eiatterminalpair 1isnumericallyidentical

edancezn;thecomplexvoltageE2atter-

yidenticalwiththetransferimpedance

anyofthese transferrelationslike

toobservethatI\ mustbethesourceand

his specificrelationisinvalidifE2 nowis

asaresponsebecausethischangeofattitude

derwhichtheparticularrelations206are

lrelations205(namely,J2is nolongerzero).

lationsmaybeextractedfromEqs.205


/ h t t p : / / w w w . h




CIRCUITEQUATIONS

istheonlynonzerocurrent,namely,

btainrelationsfor thez'slikeEqs. 207which

alinterpretation;andoncemoreit mustbe

ferrelationsapplyonlyif I2isa source.

slikeEi= znliorE2= z22/2,incon-

ofwhetherthevoltageor thecurrentisthe

onis placeduponthenonzeroJi inEqs.206

Eqs.208.A driving-pointrelationalways

ofwhichofthe quantitiesEorI isthesource

,whilethederivationofatransferrelation

ctionwhichfastenstheroles ofsourceand

esofthe twoquantitiesEand/.

pressionfor E{asgivenbyEqs. 205into

mmationinthelatter isrestrictedtothe first

£zikljk(209)

fthe twodoublesumsisseento followfrom

k=z*,\As laterdiscussionswillshow,itis

epropertiesof driving-pointandtransfer

lt.Atthis timeweshallconsideronlythe

orasingle drivingpoint(p=1) whichreads

di| h| 2(210)

193,194for T0,F0,V0,thisresult isthe

heonegivenby Eq.189topermitthe con-

frequencies.Itsusefulnessmaybeattributed

nsT0,F0,V0are realandpositiveinspite


/ h t t p : / / w w w . h




FUNCTIONS

ttheIk's mayhaveasaresultof satisfying

omplexfrequencys.Thisfactmayreadily

(213)

erealbut otherwisearbitrary.Then

bi-a,-6*)(214)

oEq.192for T0,andconsiderfirstthe

bewrittenasthe differenceoftwosums,

erchangetheletters * andk(whichwecan

ysummationindexes)andthennotethe

Lkilitbecomesclearthatthetwo sumsin

ethattheir differencevanishes.Thefirst

nt,namelythatT0 isrealforanycomplex

2 nowyieldsT0inthe form

16)

sthateachofthese doublesumsisaquadratic

einstantaneousvalueofstored magnetic

iablesaredenotedbytheletteri whilein

involverespectivelytheletters aandb

ouscurrentsik,arerealquantities.Since

atedtoapassivenetwork,itsvaluescannot

terwhatvalues(positiveornegative)are

ne canthroughtheinsertionofcurrentsources

worktohaveanysetofvalues,and yetthe

neticenergymustalwayshaveapositive

cformlike Ttohaveonlypositivevalues

of itsvariablesmaybe(it isthenreferred

adraticform)mustclearlybetheresultof its

tainrelativevalues;thatisto say,itisa

haracterizingthequadraticform.It fol-

uadraticformsinEq.216 canhaveonly

eT0canhaveonlypositivevalues.


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



CIRCUITEQUATIONS

193and194 forF0andF0are identicalin

umentshowsthatall threefunctionsT0,F(l.

anycomplexfrequencys,andthatthis

tivedefinitecharacteroftheinstantaneous

Vas givenbyEqs.Ill,112,113.

ving-pointimpedanceisobviouslynot

dependenceuponthecomplexvariablesis

nsT0,F0,V0implicitlyarefunctionsofs

eIk'swhichare solutionsoftheEqs.191for a

helesstherepresentationforznas givenby

ermineall ofthepropertiespertinenttothe

ofalinearpassivenetwork.Suchdetermina-

etodevisingmethodsofsynthesisfor pre-

ons,willbecarriedoutinthediscussions

s.

o shownextthatanalogousresultsperti-

onsareobtainedthroughfollowingaprocedure

e onejustgiven.Thusin thenodeequi-

cetheassumptions

217)

heexponentialfactor,have

inthissetwemultiplyby EitE2,E,,

ntroducingthenotation

k(219)

Ek(220)

fthetwosums ineachequationfollowsfrom

«=C*,.etc.,theresultof theseoperations

22)


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



FUNCTIONS

195obtainedonthe loopbasis,andthe

T*0arerespectivelydualto T0,F0,Vo-

are realandpositiveforall complexEt

olutionofEqs. 218forachosencomplex

ertheseequationsappropriatetoap terminal-

methatnonzerocurrentsources areapplied

rs,andwiththe abbreviation

23)

8in themoreexplicitform

n=Ip

=0

equationswenowrepresentin theparti-

artitioningusedin Eq.198exceptthathere

columnmatrices

onedasindicatedby


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



CIRCUITEQUATIONS

theEqs.224in theequivalentmatrixform

onsmaybesolvedfor [Eg]giving

hefirstequationyields

* [Ep]=W(230)

redsetofequationsinvolvingonlythe volt-

epairs.

admittancematrixoforder p

]

theequivalentalgebraicform

ftheEqs.218appropriateto thesituationin

feedingonlythefirstp nodepairs.The

uationsaredeterminedfromthe17,.* (Eq.223)

amannerindicatedbythe matrixEq.231.

8to 232maybedescribedasa processof

theinaccessibleorunwantednode-pair

re derivedspecificallyontheassumption

dtheE'sareresponses,theycorrectlyrelate

eseventhoughsomeorall oftheE'smaybe

glysomeorallofthe /'sbecomeresponses.

areregardedassources,thenthe resulting

eterminalpairsareexplicitlygivenbythese

cumstancestheterminalpairsareallshort-

ontheyn, arereferredtoasa setofshort-

ansferadmittancescharacterizingthepter-

parewiththeanalogousquantitiesdescribed

ncenetworks).


/ h t t p : / / w w w . h




FUNCTIONS

nonzero voltagesource,then

equalin valuetoonereferencevolt,then

rminalpair 1isnumericallyidenticalwith

cey\\\thecomplexcurrentI2atterminal

ticalwiththetransferadmittance2/12=J/21;

setransferrelationslikeI2 =y12#i,itis

Eimust bethesourceandI2 theresponse;

validifI2now isregardedasasourceand

his changeofattitudeviolatestheconditions

relations233areextractedfromthe general

sno longerzero).

lationsmaybeextractedfromEqs.232

s theonlynonzerovoltagesource,namely,

obtainrelationsforthe ?/'slikeEqs.234which

alinterpretation;andoncemoreitmustbe

ferrelationsapplyonlyif E2isa source.

slikeI\= ynEi\,orI2=2/22^2,in con-

ofwhetherthe voltageorthecurrentisthe

tionisplaceduponthenonzeroEi inEqs.233

Eqs.235.Asstated before,adriving-point

rdlessofwhichquantity,Eor /,isthesource

whilea transferrelationisvalidonlyfor a


/ h t t p : / / w w w . h




CIRCUITEQUATIONS

rolesof sourceandresponse,namelythatone

vation.

inpassingthat,ifthe Eqs.205and232

terminal-pairnetwork,thentheymustobvi-

he matrices[zpp]and[ypp],Eqs.204and 231,

thesematriceshavethedeterminantsZ

d Y,k,then

Z(236)

pressionfor /,asgivenbyEqs. 232into

mmationinthelatter isrestrictedtothe

k=£ yikEiEk(237)

fthe twodoublesumsisseento followfrom

* =y*,-.ForP=1 weobtainthespecific

drivingpoint.

t=y„| Ei|2(238)

s219,220,221for V*0,F*0,T*0thisresult

nofthe onegivenbyEq.188 permitting

mplexfrequency.Liketheexpression212

from thefactthatthefunctionsF*0, F*0,

esforall complexs,themethodofproof

eciselythatgiven intheconsiderationof

ionsbetweenT0,F0, Vowithandwithout

vthatapplyto s=ju.If Eqs.195and222

kwithsources atppointsof access,thenthe

gates,andhencetheleft-handsidesof these

econjugates;thatis,

*0+ T\ / i


/ h t t p : / / w w w . h




T0=rV|« I2(242)

sionsforthesefunctionswithpertinentones

er,showsthatfors= juonemaymakethe

=T*0/w2,V*0= 4FBv=V0/w2

ysicalinterpretationof thefunctionsT0,F0,

eadilypossible,butthis factisoflittle con-

mathematicalratherthantheirphysicalsig-

ducingthemintothe presentdiscussions.

utthistext theprincipleofdualityis

ofconstructinga networkdualtoagivenone

cularsituations.Itis pointedout,moreover,

ageare duals,impedanceandadmittance

s.Foragivennetworkand aselecteddriving

eofsuch apairofdual networksequalsthe

henceitfollowsthat theimpedancesofthe

al.Theproblemoffindinga networkwhose

sthereciprocalofthe impedanceofagiven

e canconstructitsdual.This andnumerous

ftheprincipleof dualitysuggesttheappro-

ts mostgeneraltermsthemethodofcon-

dofrecognizingtheir associatedreciprocal

ematrices.

uctionofadual networkdoesnotdepend

s(resistances,inductances,orcapacitances)

esin thegivennetwork.Thusthefirststep

ructingthedualgeometryaccordingto the

siongiveninArt.9 ofCh.1.Ofsignificance

vennetworkgraphbemappableonthesur-

themethodofconstructionyieldsa dual

priatelynumberedaswellasprovidedwith

ndingtothoseonthebranchesof thegiven

thegivennetworkgraphand thecorre-

raph,thematterofelementassignment


/ h t t p : / / w w w . h




CIRCUITEQUATIONS

that:AresistanceofR ohmsinonenetwork

ofRmhosintheother(or aconductanceof

resistanceofGohmsin theother);aself-

comesacapacitanceofLfarads,andacapaci-

saself-inductanceofChenrys.Thusto each

vennetworktherecorrespondsthepertinent

etwork.

networkbecomeself-inductancesinthe

tercannotbeconstructedifthe givennet-

oupledinductancesbecausethereexistsno

gbetweencapacitances.Undercertaincir-

bletofinda networkinvolvingnomutual

vingarestrictedequivalencewithrespect

es containmutuallycoupledinductances

orkof Fig.5relativetothe oneinFig.2(a)

sedin Art.5above).Insuch"equivalent"

ualinductancesarereplacedbyoneormore

emaybe representedbyadditionalbranches

network.Havingfoundsuchamutual-

t"network,onemay,tobesure,constructa

fthe "equivalent"networkisstillmap-

altotheoriginalnetworkonlyto theextent

sequivalenttoit.

m"dual"inanysituationthatinvolves

aidtobe thedualofanotheroneonly ifit

hcasethispropertyis amutualone.Wherever

espeakofthe networksasbeingreciprocal

essthenature oftherestrictionisobvious,

llythatthenetworksarereciprocalwith

example,wemighthaveapairofnetworks

pecttoa singleterminalpair,orwithrespect

atterevent,impedancematrix204forone

calwithadmittancematrix231of theother

admittance)matricesareinverse].Itmay

alofa pterminal-pairnetwork(intheabove

rdlessofmappabilityorofthe presenceof

.

themappableandmutual-inductanceless

aythinkof creatingpointsofentryoraccess

onnectingleadstothe nodesatthetwoends

ngiron"typeof entry)orbycuttingintoa

air (the"pliers"typeofentry)as pointed


/ h t t p : / / w w w . h




triesofbothtypesaremadein agiven

theoriginalstateofthat networkisleft

pairsresulting fromthe"solderingiron"

uitedwhilethosecreatedbythe"pliers"

uited.Thesetwowaysofcreatingterminal

may,therefore,bereferredtoashavingrespec-

uitcharacter;andthustheyareclearlyrecog-

ures.

networkwecreateapointof accessby

d,thenthecorrespondingentryinthe dual

the"pliers"method,andvice versa.More

ngiron"methodisappliedto theterminals

ofapairof dualnetworks,thenthepliers

orrespondinglynumberedbranchintheother

rymay,ofcourse,bemadein eachofthe

rs arethuscreatedineachofthe twodual

toncefromthe dualcharacterofthepro-

oregoingarticlesthattheset ofopen-circuit

mpedancesz,kdefinedforonenetworkby

alwiththe setofshort-circuitdriving-point

y,kdefinedfortheothernetworkbymatrix

uitimpedance(resp.short-circuitadmittance)

orksmustbemutuallyinverse.Careful

tementsisessential,andsothe following

rofdual networksinwhichppointsof

ydualcharacter)aremadeineach.Letus

A andB;andsupposewedefinefor Athe

matrix[z.*Uwithelementsz,kWandforB

cematrix[y,k]Bwithelementsy,kW-It

withindexessand kindependentlyequal

pinclusive.

pointimpedancez*k(A)attheterminal

challotherterminalpairsare open-circuited

-pointadmittanceattheterminal

challotherterminalpairsare short-circuited

ywerecreatedbythe"pliers"or bythe

fwethinkofboth ofthesedriving-point

thenclearlytheyhavereciprocalvalues;and

ving-pointimpedancesinnetworkAthat

ytopdriving-pointimpedancesinnetwork


/ h t t p : / / w w w . h




CIRCUITEQUATIONS

etworkBallterminalpairs excepttheone

to areshort-circuitedwhileinnetwork.4

ctions,wemaysimilarlysaythatanyopen-

eofnetworkAequalsthecorrespondingshort-

eofnetworkB;butitis nownotappropriate

uantitiesasimpedancessincethereciprocal

lthoughdimensionallyanimpedance)isnot

ausethereciprocalofatransferfunctionis

,ashas beenemphasizedabove.Thatisto

,forexample,isunderstoodtobethe ratio

rrentinputandthus fastenstherolesof

entandvoltagerespectively.Thereciprocal

allyanimpedance,isnotaquantitythat will

nmultipliedbya currentinput,butrather

anoutputcurrentwhendividedinto an

uallyaccomplishnothingthroughwriting

ofdivisionthatmustbe carriedoutinorder

inputundoesthereciprocalformofthis

fullythatsimilarcommentdoesnotapplyto

szkk'A)andykk'B)sincetheyareresponse

eupor upsidedown.

just aswelldetermineashort-circuit

andforthenetworkBanopen-circuitimped-

do,wewill findthattheseareidentical,and

se of[z,k]Awhile[z,ic]bistheinverseof LV.*bj.

xpairs,oneshouldnot makethemistakeof

y,kU)andz,*(il)or y,kiB)andz,*(B)have

eknowfromthedefinitionofinverse matrices,

reciprocalvalues.

gterminalpairsthroughsolderingleads to

h,orthrough cuttingintoabranch.Inthis

waysarisesastowhywecannotalso create

eringleadstoany nodepairwhetherlinked

sweristhatwecan,but that(a)itmaynot

weshouldcreatethecorrespondingentryin

he latterentrymaynotexist.Regarding

ceintothe givennetworkanadditional

rin questionand,forthenetworkgraph

ctthedual.Theaddedbranchineithernet-

zeroresistanceoraninfiniteresistance(zero


/ h t t p : / / w w w . h




whethertherespectiveentiyis theremade

olderingiron"method.Regarding(b)it

eaugmentedgraphmaynot bemappable,

no dual,andhencethereexistsnocorrespond-

l totheoriginalnetwork;thecontemplated

kis pointlessexceptforitssignificanceas

alone.

fytheuseof theterms"dualnetwork"

throughpointingoutthattherelationshipof

etworksis ageneralizedrelationshipofreci-

withthe so-calledreciprocitytheorem)

rksarereciprocalwithrespecttoallcreatable

relationshipbetweenapair ofreciprocal

sensebe regardedasasortof restricted

beregardedasbeingdual withrespectto

ryandonly withrespecttotheseterminal

rilyhavethesametotal numberofbranches

n generalitisnot possibletocreateaddi-

tsofentry.Whilethedualnetworkexists

ona sphere,areciprocalnetworkwithrespect

ointsof entryalwaysexists.*

waysasks:"Ifwecanchooseanynumber

vennonmappablenetworkandstillconstruct

oseallcreatablepointsofentry? Won'tthe

dual?"The answeris:"No,fortherecipro-

anchesthanthegiven networkandhence

ofmakingadditionalpoints ofentry,none

ndentsinthegivennetwork.Thereforethe

sbecauseduality,ashas beenemphasized

ualproperty,andthepairofreciprocalnet-

ngtoconstructdo notpossesssuchcom-

"Andthuswebringto aclosethediscussions

hodonot restrictthetermdualitywithregardto elec-

e.In fact,severalpapershavebeenwrittenshowinghow

"networkeventhoughthegivenonedoes notpossessa

ggestmoreoverthatwemustdifferentiatebetweentheuse

iedtothepurelymathematicaltheoryoflineargraphsand

Icannotsharethisview,forthe veryessenceoftheconcept

racter.Ifthereisanywayof distinguishingwhichis

ablydualnetworks,thenthesearenotduals.It ismore

chnetworksashavingspecificreciprocalproperties.


/ h t t p : / / w w w . h




CIRCUITEQUATIONS

tancevaluesare1farad each,theself-inductancesare

tualinductancesinmagnitudeare1/4henryeach.Assum-

sare allequaltoe(<)— cost,determinetheequilibrium

sisusingthemeshcurrents(with aconsistentclockwise

na nodebasis,identifyingthebranch-voltagedrops

evariablesei,«2,«s,«4.

ofthevoltagesource e,iasadriving point,determine

circuit ofProb.1,andwriteit intheformofa quotient

lexfrequencyvariables.

rcuitofProb.1,find theimpedancefunctionbetween

andputitinto theformofa quotientofpolynomials.

n

ternatelybe writteninthematrixform

m

hthe transformation

matrix[P]satisfiesthe condition


/ h t t p : / / w w w . h




rmFin Prob.4expressedasafunctionof thenewcoor-

o-calledcanonicform

n2

5deducethatFmaybe showntobeapositivedefinite

epresentation[A]=-[PJi-lP]wherein[P]is arealnon-

ssuming[P]inthetriangular form

s

<1U=P13P1* +P23P2* +PSiPik,

theP^maybecomputedfromagiven setofO,-* bya

process.Theexistenceofrealfinitepa(nonzerofor

dsufficientconditiontoprovethat Fispositivedefinite

ngular).

esaretohavetheselfand mutualvaluesgiveninthe

ysicallypossible?Isit possibleifallthe mutualin-

dystateallthe loopcurrentsinanetworkarein phase,

T,and Vvarybetweenzeroandtwicetheiraverage

ditionisalwaystrue forlosslessnetworksandhencevery

works.Asingledrivingpoint isassumed.

mEqs.191 wesetthedeterminantequalto zerowe

teristic(ordeterminantal)equationwhichdeterminesthose

naturalfrequenciesofthe network.Ifweregardthe

asingle voltage(inanyloop)asthe excitation,thenthe

ofresponsetoexcitation)allbecomeinfinitefors equalto


/ h t t p : / / w w w . h




CIRCUITEQUATIONS

chans valuethecircuitisin resonance;afinitei

glysmallexcitation.

llexcitation,thesystemofEqs. 191ishomogeneous,

iszero weknowfromthetheoryofalgebraicequations

currents,onetoanother,aredetermined.Specifically,

e toeachotherasthesuccessivecofactorsoftheelements

wby implicationreferringtotheloopin whichthevanish-

ated.Sincetheseratiosare thesameforanyrow,we

thecircuitisoperatingatresonance,thecurrentdistribu-

rkisfrozenso tospeak;thatisto say,itisthe same

excitation.

ybeessentiallytruealso forasystemoperatingneara

nordertoobservethisinterestingresult,considerthenet-

ductancesandcapacitanceshaveunitvaluesandthe

ms.Assumingtheexcitationtobea steadysinusoidalcur-

des,andaresponseto bethevoltageofanynodewith

putevaluesofalldriving-pointand transferfunctionsfor

rkofthepreviousproblemwithnodes3 andcjoined.

ts ofProb.9andthe factthatatresonance=V,v,

— uo'To(for^1=1) isnearlyconstantforthevicinityof

cefrequencyuo.Thusobtainforthedriving-pointimped-

roximateexpression

emintheusualmanneras Q=ao/wwherewis thewidth

tweenitshalf-powerpoints,showthatoneobtains

problembyconsideringthecorrespondingdualrelation-

eros andpolesthataremuchcloserto thejaxisthan they

eapproximaterepresentation

P^/Tv,aretobe evaluatedatthefrequencyinquestion

traintsatthedrivingpoint.

etworkisexcitedbya voltagesourceinloop1.The

energyfunctionsare

=iiJ -4t'i*2+2tit,

jj3

edto thetransformation


/ h t t p : / / w w w . h




nd Tintermsof i''i,i't,t's,and constructasimplenetwork

ables.Showthatthis networkhasthesamedriving-point

one,andhencethatthe abovemanipulationsconstitutea

erealizationofthisadmittance.Discussthelimitations

alsynthesismethod.

ceandinductancematricespertinenttotheoriginal net-

em,determinethedriving-pointadmittanceforloop1as

andfromits partialfractionexpansionobtainbyinspec-

hat foundabove.Contrastthissynthesisprocedurewith

etworkis operatingverynearlyatoneofits resonance

gthevaluesoftheenergyfunctionsTavandFav.Whatis

edance?WhatistheQofthe networkforthisresonance?

os\/3tthe circuitofProb.15isoperatingin thevi-

cefrequency.AgaincomputeTtY,Vnv,andQ.What

esbeinorder thattheresonancefrequenciesmayfallat

ndtheQ'sremainthesame?

ceofthecircuitofProb.15 acrossthenodepairsa-b,

du=\/3to showthattheseareresonancefrequencies

heseimpedances.Computetheimpedanceforacutatm

sonancefrequenciesnowlienear thezerosofthisim-

wsacircuitconsistingoftwo inductanceswoundona

reandtwo1-faradcapacitances.Theself-inductancevalues

itudeofthemutualinductanceis1/2henry.


/ h t t p : / / w w w . h




ymmetricallatticeshowninthe sketchbelowarein

woterminal-pairnetworkwhosez matrixistheinverse

vennetwork.Evaluatebothzmatricesandcheckthat

ically,whatisthedualof thelatticestructure?

Prob.23to thesymmetricalbridged-teenetworkshown

ainareinhenrysand farads.


/ h t t p : / / w w w . h




/ h t t p : / / w w w . h




96

Impedance

155,369,

definitions,54,

36

8

alculationof,

90

35,316

5,316

,5

es,305

230

ynetwork),419

412

cy,421

soids,273

orks,307

urrent),89

0

cuit,247

,70

48

n,127

0

lution,116

uilibrium,234,

cy-domain

433

,452

21

pts,43

535

344,512

2

4,523


/ h t t p : / / w w w . h




69

412

38

36

99

ng-pointand

7

quency,412

s,315

9

tationof,

s,354,523,

1 ,

ntary,305

alculationof,

89

quilibri<

consistency

1,385

0

69

es,297

matrices


/ h t t p : / / w w w . h




71,491,498,

1

163

48,518

tion,431

337,454

s,356,542

74

3

parts,303

05

,369,429

ms,280

culation,211

88

ce,301

esponse

229

370,462;see

352

equency,309,

s155

,203

37


G o o g l e - d

i g i t i z e d

/ h t t p : / / w w w . h


_ u s e # p d - g

o o g l e



38

introductory circuit theory by guillemin ernst

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