D DAVID PUBLISHING. Filling the Teaching Gap between Electromagnetics and Circuits. Nomenclature. 1. Introduction. Massimo Ceraolo

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1 Joural of Elecrical Egieerig 5 (07) 3-4 doi: 0.765/38-3/ D DVD PULSHNG Fillig he Teachig Gap bewee Elecromageics ad Circuis Massimo Ceraolo Uiversiy of Pisa, Pisa, aly bsrac: Elecrical egieers ormally are augh elecromageism i a elecromageics course (e.g. i [-]), ad circui aalysis i a idepede course (e.g. i [4-6]). Circuis are domiaed by Kirchhoff s laws, while elecromageics is domiaed by Maxwell s equaios. However, he correspodece bewee wo ses of equaios is o immediaely perceived ad his creaes some uceraiy i he youg elecrical or elecroic egieer, which may grow wih he doub ha Kirchhoff s laws may be somewha laws of he aure idepede of he laws of elecromageism. This paper has he purpose of supplyig eachig maerial ha may be used o fill he gap, ad herefore be augh eiher a he ed of a elecromageics or a he begiig of a circui course. explois large pars of he paper published i a coferece [8], bu also coais sigifica ehacemes. The paper firs shows simple disribued parameer sysems, whose behaviour follows Maxwell s equaios, ad he shows ha hey, uder give assumpios, ca be modelled as circuis, whose behaviour is govered by Kirchhoff s laws. Key words: Circui aalysis, Kirchhoff s equaios, Maxwell s equaios, eachig. Nomeclaure mageic flux desiy (or mageic iducio) D elecric flux desiy E elecric field J curre desiy field (or areic curre field) displaceme curre desiy field (or areic J d displaceme curre field) H mageic field dielecric cosa elecric resisiviy µ mageic permeabiliy flux likage c charge desiy (or volumic charge) C or a.c. DC or d.c. EMF KVL KCL. roducio aleraig curre direc curre elecromoiveforce Kirchhoff s Volage Law Kirchhoff s Curre Law Durig pas ceuries he elecromageism heory has see he basic laws firs (such as Gauss s, mpère s, Faraday s, Ohm s) o be discovered i a Correspodig auhor: Massimo Ceraolo, MS i elecrical egieerig, full professor i elecric power sysems, research fields: circui basics, elecric ad hybrid vehicles. iegral, macroscopic way, he o be expressed i a differeial form ha resuls i equaios ha, while havig as a cosequece he iegral versios from which hey derive, are useful exesios of hem. The mos impora effor i his raioalizaio of he basic laws was from Maxwell, ad herefore he resulig equaios are called Maxwell s equaios (i differeial form). Raher idepedely, he basic circui laws, kow as Kirchhoff s Curre Law ad Kirchhoff s Volage Law, have bee posulaed ad widely used. Kirchhoff s laws are augh o be applicable o circuis, which ca be fuzzily defied as elecromageic sysems composed by lumped compoes coeced o each oher by hi coducor lies (= wires). The wo equaio ses (Kirchhoff s ad Maxwell s) however, are idepedely iroduced, so i is o always clear wha is he raioale behid he posulaio of Kirchhoff s laws i circuis, or, equivalely, wha are he hypoheses ha allow a physical, hree-dimesioal, sysem o be modeled ad sudied as a circui (govered by Kirchhoff s laws)

2 3 Fillig he Teachig Gap bewee Elecromageics ad Circuis [-7]. Ref. [8] was herefore coceived wih he purpose of fillig he kowledge gap bewee elecro-mageics ad circui heory, so ha he relaio of he wo approaches is clarified. This paper reproduces pars of he resuls of Ref. [8], havig i mid he eed o ehace, akig advaage of wha was here discussed, he way eachers each he fudameals o which circuis, as a cocep, are based.. Graphical Coveios To faciliae udersadig he logical disicio we wa o make bewee elecromageic sysems havig a circuial shape (i.e. beig cosiued by lumped compoes coeced o each oher by hi lies), we make a graphical disicio: i case of physical sysems we reproduce he cross-secioal size of he lies, while i case of circuis we do o (Fig. ).. Maxwell s ad Oher Releva Elecromageism Equaios lhough very well kow, here he four Maxwell s equaios are reproduced i heir iegral form, so ha hey cosiue a easy referece whe readig of he remaider of he paper. The symbols are hose repored i Secio, ad used hroughou he paper. l( S ) l( S ) S ( v) Edl DdS H dl ds c dv ( J J ds 0 closed S S S ( l) v( S ) S ( l) d ) ds; J d () dd d addiio o Maxwell s equaios he poiwise Ohm s ad coiuiy equaios are remided, because of heir imporace for he paper. E J () J ds ( ) c dv (3) S v v( S ) case of sysems where i is kow ha all quaiies are cosa (i.e. DC sysems), all ime derivaives become zero, ad he firs ad hird Maxwell s equaio become: Edl 0 closed l l (4) H dl ( ) J ds l S S ( l) ad coiuiy equaio becomes as well: S J ds 0 closed S (5). Disribued Sysems ad Circuis Elecromageism sudies elecrosaic (i.e. relaed o effecs of he presece of charge i give porios of space) ad elecrodyamic, or mageic (i.e. relaed o movig charges) pheomea. exeds o pheomea relaed o ieracio of he previous wo, sice i ime-varyig sysems hey are closely relaed o each oher. The use of Maxwell s equaios or oher elecromageism ools eiher i iegral or differeial form has prove oo demadig o aalyze sysems composed by differe elecromageic subsysems. Cosider he simple sysem show i Fig.. is formed by a elecric sie-wave volage geeraor feedig wo lamps wih he ierposiio of a couple of wires, which are represeed hick, because i a physical sysem hey have o oly a legh bu also a widh. G device device device a) b) device Fig. simple elecromageic sysem coaiig wires coecig lumped compoes. Fig. Graphical coveio adoped for coducors: (a) devices coeced by real-coecig lies; (b) devices coeced by idealized lies. The coiuiy equaio is o idepede of Maxwell s equaios: i ca be easily derived akig he divergece of boh members of he hird of (), ad iroducig c i he resul akig i from he secod oe.

3 Fillig he Teachig Gap bewee Elecromageics ad Circuis 33 The aalysis of his sysem would be grealy simplified if, isead of havig o aalyze simulaeously he whole sysem usig Maxwell s equaios (differeial equaios o be applied a ay poi of space akig io accou all boudary codiios), we are able o wrie idepede equaios of he ivolved idividual, lumped, compoes ad lik hem by some addiioal cogruece equaios. This approach ca be referred o, for he ime beig, as he circui or circuial approach. qualiaive aalysis of Fig. shows ha he geeraor coecs o he lamps hrough log wires, while shor, verical coecios are prese a he wo sides of he sysem. Therefore a hypoheical approximaio of he sysem of Fig. could be as show i Fig. 3a: he coecios are show usig hi wires, o evidece heir coecig role, while he pars of he origial sysem o be modeled idividually are eclosed i boxes or circles. Fig. 3b, a furher evoluio of he sysem is show, i which he compoes are subsiued wih symbols idicaig specific mahemaical modelig of he cosidered compoes: ideal resisors for lie ad loads, ideal geeraor for he geeraor. The circuial approximaios of Fig. 3 are composed oly by circui elemes (geeraor, lamps or resisors, rasmissio lie box) ad ideal wires. ll physiciss ad elecrical egieers already kow very well ha his lumpizaio of elecromageic sysems cosiued by compoes joied by coducor wires is possible, bu rarely he raioale behid his coversio is ivesigaed. he followig secios i is show ha he coversio of spaially-disribued physical sysems io circuis is possible, uder cerai hypoheses, which also deermie he choice o how o make he rasformaio, ad imply some limiaios. search of he implemeaio of he coversio io circui of ay sysem govered by he elecromageism equaios, beer is o sar wih he simples case, i.e. whe all quaiies do o vary wih ime. y radiioal omeclaure hese sysems are referred o as direc curre sysems. 3. pplicabiliy of Kirchhoff s Curre Law i d.c. Circuis Cosider a regio of space, able o exchage charge bewee is ierior ad exerior. case we wa o aalyze a sysem by meas of he echique of coversio io a circui, i is raher obvious ha his charge exchage occurs oly by meas of discree chaels cosiued by he wirig eerig he surface, while charge exchage i regios o occupied by wires is egleced. Therefore i is reasoable o pu forward he followig: SSUMPTON : ay charge flow is egleced aywhere ouside circui elemes, excep ha wihi coducor wires. Cosider he regio of space V surrouded by surface S (Fig. 4a). he drawigs of Fig. 4, all he coducor wires covergig io he volume V are cosidered (ad show). G a) b) Fig. 3 Circuial approximaio of sysem of Fig.. J J S Trasmissio lie S J 5 V S 5 S S 4 J S 3 J 4 J 3 a) s 5 4 Fig. 4 cofied regio of space (a) ha ca brig o a geeralized ode (b) ad a ode (c). eer ame would have bee cosa operaio sysems or seady-sae sysems. 3 5 c) 4 3 b)

4 34 Fillig he Teachig Gap bewee Elecromageics ad Circuis Le us ow cosider he coiuiy Eq. (3). J ds ( ) c dv. S v v( S ) where c is he spaial charge desiy, or charge per ui of volume. Sice by hypohesis we are i d.c., meaig ha i he cosidered sysems all quaiies are cosa, charge desiy is also cosa ad herefore i his case he coiuiy equaio becomes Eq. (5). ecause of ssumpio coducive curres are possible oly wihi wires. So he iegral of Eq. (5) is simplified sice J is o-zero oly hough S k, ha are he iersecios of S wih coducor wires. S J ds where k Thus: k S k J ds S k k k J ds 0 k ( k ) 0 Tha is KCL for he regio V. Fig. 4b shows aoher represeaio of he same sysem of Fig. 4a. his paper, however, here is a logical disicio bewee he wo: he hi wires of Fig. 4b are idealized wires, based o special assumpios. his poi of he paper, he oly assumpio beyod he symbol of idealized wires is ssumpio. Obviously eough, he demosraio proposed, referred o he scheme of Fig. 4a, coaied wha is ormally called i circui ermiology a geeralized ode, is applicable also i he scheme of Fig. 4c coaiig a coveioal ode: i is jus ecessary o cosider a iy surface S aroud he coecio of some wires. CONCLUSON : i a physical sysem operaig i DC for which ssumpio is applicable he KCL applies. 4. pplicabiliy Kirchhoff s Volage Law i d.c. Circuis Cosider he sysem displayed i Fig. 5. E c _ b _ a Fig. 5 asic sysem wih charge flow ad chemical ad elecrosaic fields. is composed by a elecrochemical baery (a he lef side of he figure) coeced hrough physical wires o a load resisor R l a he righ side. Le us firs imagie ha here is some posiive charge locaed a he upper ermial of he baery ad a equal amou of egaive charge i he lower oe. These charges would creae a elecric field i he space aroud hem: iside coducors i is logiudial, while ouside i has a differe orieaio (oe possible force lie is show dashed i figure), bu has o relevace for aalysis of DC sysems sice by effec of ssumpio, charge moveme is allowed oly wihi coducor wires. y idividual charge prese i he coducors (i.e. a elecro) would he circulae i he coducor loop of he sysem, ad fially offse he iiial charge accumulaed a he wo baery ermials, ad so i a very shor ime he coducor loop would be eural ad o more charge could flow. Durig his flow, he eergy received by he charge by effec of is dissipaed durig he rasi, by effec of he eergy loss occurrig durig charge moveme i coducor maerials, i.e. where Ohm s law applies. is a well-kow fac ha he baery is able o cause coiuous charge flow i he circui. lhough he acual behavior of a elecrochemical baery is very complex, i ca be modeled for he purposes of his sudy as a sysem able o pump charges pushig hem from is egaive elecrode owards is posiive oe by meas of a ier elecric chemical field E c ha his way les charges flow. y charge loopig i R l

5 Fillig he Teachig Gap bewee Elecromageics ad Circuis 35 he sysem of Fig. 5, obais eergy whe i goes hrough he baery, by combied effec of E c ad, which is laer ad delivered (dissipaed) whe he charge flows hrough he exeral circui. The eergy supplied by E c o he charge equals he eergy dissipaed durig he flow. Sice i he eire loop E c has a e coribuio o he work rasferred o he charge, i is a o-coservaive field. Therefore aalysis of his circui ca be made sarig from he supposed simulaeous presece of elecrosaic, coservaive ( ) ad chemically iduced, o-coservaive (E c ) fields i he baery: E = E c (6) The charge moveme i he loop is deermied by he presece of he whole field E, o oly ; herefore he Ohm s law is o be wrie: E J ad, akig he loop iegral of boh sides: E dl The lef par of Eq. (7) is: E dl J dl (7) b a Ec dl (8) because of he coservaiviy of ad he absece of E c ouside he baery. The iegral of righ par of Eq. (7) may be compued eglecig he resisace of coducor wires i compariso o baery ad load resisaces. b a J dl a b J dl J dl J dl a J dl ( R Rl ) J dl (9) while he laer equaliy is jusified by he relaios, for boh resisive compoes: dl J dl ( l) J ( l)dl ( l) dl R S( l) S( l) where i has bee exploied ha, as a cosequece of he coiuiy equaio, curre does o deped o he iegral variable l. Subsiuig Eqs. (8) ad (9) io Eq. (7) gives: Ve E c dl ( R Rl ) (0) where he quaiy V e, defied by meas of E c, is called elecromoive force (EMF) of he circui (subscrip e sads for elecromoive) 3. Eq. (0) is a usual expressio of Ohm s law for oe-loop sysem, ad may be cosidered o be he resul of applicaio of KVL o he circui of Fig. 6, ha assumes he role of equivale circui of sysem of Fig. 5. The uilizaio of KVL i his circui is ow validaed by meas of he Maxwell s ad Ohm s equaios. lhough raher obvious, i is impora o sress ha he resul obaied is o jus liked o he presece of a elecrochemical baery. Several possibiliies exis o creae devices ha i is iside pump chages from is egaively charged ermial o is posiively charged oe, i.e. hey make charges move hrough he exeral circui hrough he elecric field creaed i he coducor by he charges locaed a he ermials of he pumpig devices. 4 Le us ow cosider a more complicaed elecric sysem show i Fig. 7. coais several loops, resisors ad several baeries. Moreover, i is o elecrically isolaed from he ouside world: because of he coecios a he corers of is loops. V e R Fig. 6 Equivale circui of he sysem show i Fig The erm elecromoive force is maiaied for is worldwide use; i is however appare ha i is parially cofusig, sice he quaiy o which i refers is o a force i he physical sese. 4 Sigifica sources of cosa elecromoive force are fuel cells, phoovolaic cells; elecric machies operaig as source are souces of ime varyig elecromoive forces. b a R ec

6 36 Fillig he Teachig Gap bewee Elecromageics ad Circuis he whole sysem cosidered, icludig he pars o show i figure, here exiss i priciple a field caused by he charge accumulaed a all he baery ermials (cosiderig also hose ouside he show par of he sysem), ad he correspodig curre desiy field J =, where, obviously, ay poi of space has, J, of is ow. However, because of ssumpio, here is o ieres i cosiderig he fields prese ouside he coducor wires. The direcio of elecric ad curre desiy fields iside he coducor is parallel o heir axes, bu is orieaio is o kow a priori (his lack of kowledge has preveed he possibiliy of reporig he vecor arrows i Fig. 7). Fig. 7 hree possible loops may be cosidered: L, L ad L3. Le us cocerae, wihou loss of geeraliy, o loop L. Whe a geeric charge q goes alog loop L i he sysem show i Fig. 7, he work of he elecrosaic field o i is ull, because of he coservaive aure of he elecrosaic field. Cosequely, he aalysis carried ou wih sysem of Fig. 5 ca be repeaed for ay loop of Fig 7. The Ohm s law gives: The lef par of Eq. () is: E L dl () (3) where ad ca boh be compued as J ds usig S ay cross secio of he sysem braches coaiig R ad R respecively. The above aalysis perfecly replicaes ha made for he simpler case of Fig. 5; herefore if he followig defiiios are adoped: L J dl E () dl Ec dl Ec dl L a a because is coservaive. gai, he iegral of righ par of Eq. () may be compued eglecig he resisace of coducor wires i compariso o baery ad load resisaces. L b J dl ( R R ) ( R R ) b E c R b a L L3 E c a b R L Fig. 7 muliple-loop, muliple source, muliple resisor sysem. V e V e he loop L of he sysem of Fig. 7 ca be sudied as repored i he upper par of Fig. 8, ad he eire sysem of Fig. 7 ca be sudied usig he equivale circui show i he boom par of Fig. 8. R b a b a E V e E c c dl dl - V R b a b a R - E E V e V e c c R dl dl V e R - V e3 V R Fig. 8 Equivale of loop L of he sysem show i Fig. 7 (op) ad of he full sysem of Fig. 7 (boom). - V e4

7 Fillig he Teachig Gap bewee Elecromageics ad Circuis 37 - V 5 - V V V34 - V3 - Fig. 9 Oe-loop, muliple-resisor circui. his poi a more geeral resul is obvious: i ay loop of ay DC circui he sum of all elecromoive forces (iegrals of ier, o (4) coservaive fields) equals he sum of all resisaces muliplied by he correspode curres. lhough i DC sysems elecromoive forces have a very differe physical ierpreaio ha volages across resisors, as show above, Saeme (4) ca be expressed i a more geeralform. if ay loop is goe hrough i a clockwise direcio (or, equivalely i couer clockwise direcio) he (5) sum of all volages rises (or, equivalely, he sum of all volage drops) is zero. Saeme (5) is he well kow KVL. lhough i has bee derived cosiderig he fields iside he coducors, i assumes a form ha is immediaely usable i circuis, o sae useful relaios bewee circui quaiies. The KVL ca be see o be he circuial versio of coservaiviy of elecrosaic field. deed i saes ha ay loop implies ull volage sum, which is he equivale of he oio ha ay circui iegral of elemeary work of elecrosaic field is ull. Furhermore i is also equivale o sae ha i ay circui is possible o defie a defiie volage value for ay ode, a fac ha agai recalls he correspode characerisic of elecrosaic (or ay coservaive field). The equivalece of KVL ad possibiliy o defie give poeials o circui odes ca be easily show cosiderig Fig. 9. Here differe lumped compoes (havig arbirary ier behavior) are coeced o each oher i a loop isered i a larger circui (i he mos geeral case ay ode may have a wire coeced wih i ad wih oher elecric circui compoes). The volages across heir ermials are amed afer he ermials hemselves, ad hey are cosidered accordig o he polariy refereces (posiive volage rises) show i Fig. 9. f a volage ca be defied for each ode, he, cosiderig ha he ()h ode is ideed he -s ode, i is: k V k, k Vk Vk ) k ( V V V 0 k k k Le us summarize he procedure followed i his paragraph: has bee see ha he curre coiuously flowig i a simple DC circui is cosequece of he presece of a o-coservaive field i oe or more forcig compoe; as a cosequece of his field, he ermials of he compoe wherei his field is prese are able o remai differely charged; The charge differece a he ermials of a forcig compoe causes he presece of elecrosaic field i he coducors ouside i, whose poeial differece a hese ermials is equal o is elecromoive force; This elecrosaic field is obviously coservaive ad so volages across circui ermials are idepede of he pah cosidered, ad hus a poeial ca be defied for each ode; The presece of ode poeials is ecessary ad sufficie codiio for he KVLs. The gis of his process is: forcig compoes o deermie a elecrosaic (coservaive) field i he coducors ouside hem, so poeials of idividual circui pois ca be defied ad herefore he KVL applies. This works i ay case, if we cosider ha he circuis are oally separaed from he ouside world, k

8 38 Fillig he Teachig Gap bewee Elecromageics ad Circuis excep from ieracios ha may occur iside circui elemes (such as i he baery of Fig. 7) Therefore he followig is pu forward: SSUMPTON : ay ieracio of he cosidered physical circui wih he ouside world is made oly iside circui elemes. No ieracio wih wires ad space bewee wires is supposed o occur. The, he followig coclusios ca be made. CONCLUSON : i a physical sysem operaig i DC for which ssumpio is applicable, he KVL applies. CONCLUSON 3: i a physical sysem operaig i DC for which ssumpios ad are applicable, he KCL ad KVL apply, ad herefore i ca be sudied as a circui. y ssumpios ad 3 coducive curres are possible oly wihi wires ad displaceme curres oly hrough capacior armaures. So he iegral of Eq. (5) is simplified sice oly hough surfaces S k J is o-zero, where S k are he iersecio of S wih coducor wires or capaciors. S J ds k S k J ds k 0 Tha is he KCL. The cocep is exemplified i Fig. 0, where he sums are o be performed wih k goig from o 6 (lef-side circui) or o 7 (righ-side circui). CONCLUSON 4: i a physical sysem for which ssumpios ad 3 are applicable, he KCL applies. k 5. Exesio of Kirchhoff s Laws o Time-Variable Circuis 6 7 he previous paragraphs i was see ha i circuis operaig wih cosa quaiies, ha by radiio are called DC circuis, he KVL is a direc cosequece of he elecrosaic field prese i he circui, ad KCL is a cosequece of coiuiy equaio. his secio, hese coceps will be expaded o cover also ime varyig circuis. 5. Exesio of Kirchhoff s Curre Law Whe we draw circuis we imagie ha o curre ca flow bewee circui wires. oher words, circuis have o real wires, bu ideal oes, which are such ha curre (eiher coducive or displaceme curre) ca flow oly iside wires, while cao i he free space aroud hem. Naurally displaceme curres ca flow iside circui elemes, e.g. capaciors. This jusifies makig he followig assumpio. SSUMPTON 3: he displaceme curre D / is egleced aywhere, ouside circui elemes. This is idepede from ssumpio ha referred o coducio curres, i.e. curres of he oly ype prese i DC circuis Fig. 0 Sample surfaces ad showig coiuiy equaio ad KCL i a sample circui. 5. ssues wih Log Lies: Meacircuis Le us ow ry o use he heory iroduced up o kow i a sysem composed by hree subsysems: wo of hem have a couple of ermials ha are he oly way o ierchage coducive curre wih he ouside world, ad do o allow exchage of displaceme curres (hese will be called lumped compoes). These are coeced o a disribued sysem, wherei coducive ad displaceme curre circulae. To fix ideas, le he wo lumped compoes be a geeraor of siusoidal EMF (jus as he oe used i he firs example) ad a ohmic resisor, while he disribued sysem is a rasmissio lie cosiued by wo coducors ad he surroudig space (op of Fig. ). 6

9 Fillig he Teachig Gap bewee Elecromageics ad Circuis 39 a sysem of his kid i may be uaccepable o disregard he effecs of he mageic field hrough he loop creaed by he hree subsysems; very ofe, i is also o possible o disregard he effec of displaceme curres bewee wires. The issue is complex ad requires more space ha jus a paper paragraph. raher horough aalysis is show i ppedix of Ref. [7]. Here, jus o clarify he issue, a simplified discussio is proposed, i which we eglec he effecs of displaceme curres. We jus meio here ha his gives accepable resuls for 50/60 Hz power lies ypically for lies havig leghs up o a few es of km. his example, where he rasmissio lie has a much greaer legh ha he disace bewee he wo coducors, he mageic field i he loop ca be assumed as beig equal i shape o he mageic field creaed by wo idefiie legh wires. ca also be assumed ha he effecs of he mageic field i space bewee he rasmissio lie ad he lumped compoes ca be egleced. Fially, we assume ha he mageic iducio bewee he wo coducors is due oly o he curre flowig i he coducors hemselves. Uder hese hypoheses, ad eglecig displaceme curres, he behavior of he rasmissio lie ca be described usig equaios ha do o ivolve kowledge of wha happes ouside i, ad herefore ca be subsiued by a lumped compoe (block L i boom-lef par of Fig. ). The par of he circui iside L is govered by he firs of (), ad is iegral cosequece, he Faraday s law; herefore he sysem of Fig. is described by: v ( R R ) i( ) v ( ) (6) where, v d di ( ) L d d L ad L, self-iducace of he circui, is he proporioaliy coefficie bewee curre ad he flux likage i creaes. Raher obvious, Eq. (6) ca be ierpreed as he KVL of he circui repored i he boom-righ par of Fig., from which, he, i() ca be deermied whe v() is kow, ad vice-versa. is impora o sress ha we jus demosraed how o evaluae v, or, if we wa, v vu Ri( ) v( ), while we did o meio oher volages. deed he boom-righ of Fig. cao be used o evaluae of oher volages, such as for example v D () or v C (). This because Faraday s EMF of loops coaiig simulaeously a poi from he couple C-D ad oe from he couple - does deped o he loop geomery, which i ur is a cosequece of he mageic field is o-coservaive. This ca be visualized clearly cosiderig for example a measurig sysem of volage v D i he physical sysem (Fig. ). is appare ha ay chage of he posiio of a volmeer ha would be ieded o measure v D would chage he area of he loop composed by he coducor D ad he measurig wire, ad herefore he elecromoive force geeraed accordig o Faraday s law (firs Eq. of ()). More explicily, if we cosider he wo coours c (D--v -D) ad c (D- -v -D) we ge applyig (firs Eq. of ()): d ( c) E dl d v RDi c ( c ) d c d ( E dl d v RDi ( c ) d d( v v D v() C v() L ( c) D C Lumped compoe modelig d i() ( c) ) Physical sysem R u v v v() i() D C R L c) L R u Equivale meacircui Fig. Physical sysems coaiig log lies ca be represeed by a lumped compoe model or a meacircui. R u v u -

10 40 Fillig he Teachig Gap bewee Elecromageics ad Circuis v() D i() v - V - C Fig. Differe measurig loops creae differe measures o he volmeer V. We ca draw he followig coclusios from he example. Sice circuis have he characerisic ha all volages bewee odes ca be compued, i is o possible o deermie a circui compleely equivale o a sysem havig log lies; Somehig similar o a circui ca be deermied ha has a behavior equivale o he give sysem, whe used oly for separae deermiaio of elecrical quaiies i he wo eds of he give sysem. We call his modified versio of he circui cocep meacircui. Sice meacircuis are o he same hig o circuis, a specific graphic represeaio is advisable. Tha jusifies he presece of he curved-dashed lies used i he boom-righ par of Fig., illusraig a meacircui. 6. Relaioship wih Elecroquasisaics ad Mageoquasisaics The lik bewee elecromageics ad circuis has some coecios wih oher approximaios of elecromageism (Maxwell s) equaios usually cosidered [9]: Elecroquasisaic approximaio: i cosiders he variaio sufficiely slow ha he secod erm i he firs of () o be zero. This implies E beig coservaive sice is work o ay loop is zero. Mageoquasisaic approximaio: i ivolves cosiderig he D variaio sufficiely slow ha is ime derivaive is se o zero i he hird of (). The relaioships of hese approximaios o wha is v V R u doe whe circuis are used are o sraighforward. However we ca observe ha: he elecroquasisaic approximaio meas eglecig Faraday s iducio law. We do his i rue circuis, ouside lumped compoes (i.e. bewee wires), bu we do o i wha we called meacircuis; he mageoquasisaic approximaio meas eglecig he effecs of displaceme curres, which is wha we do i his paper i circuis i he empy space bewee wires. This is o sufficie i some cases, for isace for log power lies. This special case was o deal wih i his paper, bu aalyzed i ppedix of Ref. [7]. 7. Summary We ca resume wha has bee obaied i his paper as follows. ssumpios ad allow physical sysem operaig i DC o be reaed as circui, for which KCL ad KVL are assumed o be valid. ssumpios o 4 allow a physical sysem o be reaed as circui, for which KCL ad KVL are assumed o be valid. Therefore we ca ow say ha: uder precisely saed assumpios, a sysem, which is composed by circui elemes ad coducig wires, ca usually be aalysed by meas of he mahemaical-graphical ool called circui. For hem KCL ad KVL are posulaed o be valid (Circui elemes are subsysems ha have elecrical ieracio o he res of he sysem oly hrough heir ermials). However, sysems coaiig log lies cao be reaed as rue circuis. For hem we iroduced a ew cocep, called, meacircui. ll elecric egieers kow ha whe sysems coai log lies, oly equaios relaig quaiies a each of is wo eds o each oher ca be compued, ad o cross-quaiies such as cross-volages. However, his disicio is always fuzzy. deed his paper, as well as Refs. [7] ad [8] have show ha i is very impora, ad deserves a specific ame ad specific

11 Fillig he Teachig Gap bewee Elecromageics ad Circuis 4 graphic represeaio. The approach described ca be exeded o oher sysems such as hose coaiig rasformers, elecric machies, mulipoles, ec.; however such a comprehesive aalysis is ou of he scope of his paper. The geeral approach preseed i he paper has bee adoped i book [8]. 8. Coclusios This paper had he purpose of clarifyig wha circuis are, makig a ea disicio bewee physical sysems wih log wires, which are hree-dimesioal sysems govered by Maxwell s equaios (we called hem circuial sysems), ad circuis, which are absrac graphical-mahemaical eiies, ad are very easily reaed usig Kirchhoff s laws. Kirchhoff s laws are o jus a cosequece of elecromageics laws: o use hem i subsiuio o elecromageic laws, we eed o add o hem a few assumpios, a ask ha is ormally o performed i exbooks. To give a coribuio o clarify how circuis relae o physical sysems, his paper firs saes clearly ha wha we call circuis are a mahemaical-graphical absracio of physical sysems havig a circuial shape. The i shows which assumpios we eed o add o he basic elecromageics laws o allow circuis o describe physical elecromageics sysems, hus o use Kirchhoff s laws o aalyze hem. The paper cosiders boh saioary (DC) ad ime-varyig (C) circuis. shows a impora limiaio of C circuis, which is overcome iroducig he cocep of meacircuis. The approach proposed is clarifyig for egieers ad useful for eachig, ad has bee adoped i book [8]. Refereces [] Ulaby, F. Fudameals of pplied Elecromageics. Upper Saddle River: Preice Hall, SN [] Plous, M.. pplied Elecromageics. New York: Mc Graw-Hill, SN [3] lexader, C., ad Sadiku, M. Fudameals of Elecric Circuis. New York: Mc Graw-Hill, SN [4] Hay, W. H., ad Kemmersly, J. Egieerig Circui alysis. New York: Mc Graw-Hill, SN [5] Paul, C. R., Nasar, S.., ad Uewehr, L. E. roducio o Elecrical Egieerig. New York: Mc Graw-Hill c, SN X. [6] Sarma, M. S. roducio o Elecrical Egieerig. Oxford Uiversiy Press, SN [7] Ceraolo, M., ad Poli, D. Fudameals of Elecric power Egieerig. EEE/Wiley, SN [8] Ceraolo, M. The Elecromageics Foudaio of Circuis Revisied. Prese a he 04 eraioal Coferece o Circuis, Sysems ad Sigal Processig, S. Peersburg, 3-5 Sep. 04, hp:// CCS/CCCS-00.pdf. SN: [9] Haus, H.., ad Melcher, J. R Elecromageic Fields ad Eergy. Eglewood Cliffs, NJ: Preice-Hall, SN: , Chapers 7 ad 8. hps://ocw.mi.edu/resources/res-6-00-elecromageicfields-ad-eergy-sprig-008/.