2. Electric Circuit Theory

Transcription

1 . Elecrc rcu Theory J Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna Elecrc crcu heory and Elecromagnec heory are he wo fundamenal heores upon whch all branches of elecrcal engneerng are based. Many of hese branches, such as producon, ransmsson and ulzaon of elecrc power, elecrc machnes, conrol, elecroncs communcaon, and nsrumenaon, are based on elecrc crcu heory. In elecrcal engneerng, we wan o ransfer elecrc sgnals or elecrc power from one pon o anoher. To do hs requres an nerconnecon of elecrcal deces. Such nerconnecon s referred o as an elecrc crcu, and each componen of he crcu s known as crcual elemen. Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna

2 Elecrc rcu Theory The erm elecrc crcu ndcaes he physcal place where elecromagnec phenomena are locaed. The oher meanng of he erm elecrc crcu regards he mahemacal models whch descrbes hem. Usually he erm s ulzed o ndcae he crcus and he relae models, ha sasfy he assumpon of lumped componen models (also called lumped elemen models or lumped parameer model). Ths assumpon consders all elecromagnec phenomena concenraed nsde dscree bodes (named lumped componens, or crcu componens, or also crcu elemens) whch are elecrcally conneced so ha he elecrc charges can moe beween hem. Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 3 Elecrc rcu Theory In he fgure a represenaon of a lumped model, or crcu model, made up of a olage source and a ressor, s shown. The charges are moed by he olage of he olage source and flows from he olage source o he elecrcal ressance gong hrough. Ths s possble due o he elecrcal connecons connecng he wo elemens so ha he elecrc charges can moe beween hem. - V s Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna

3 Elecrc rcu Theory The elecrcal quanes whch descrbe he elecrcal behaor of a crcu are macroscopc negral quanes (lumped parameers ha are olages and currens). They depend on me bu do no depend on space. The equaons of he crcu models are algebrac equaons or negro-dfferenal me dependen equaons ha n many cases can be reduced o algebrac equaons. The quanes of he dsrbued parameer models are usually me and space dependen quanes. The equaons of hese models are paral dfferenal equaons n me and space. Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 5 rcus n Naure and n Technologcal Applcaons Neurons and her connecons Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 6

4 Elecrc rcu Theory q In 958 he frs chp was realzed by J. S. lar Klby (Fg. A, chp s an elecronc dece ha conans seeral sold sae crcu elemens). q In 96 he frs monolhc chp (Fg. ) was made. The realzaon of monolhc crcus wh seeral crcu elemens (negraed crcus) allowed o reduce he dmensons of he elecrc and he elecronc deces followed by a rapd deelopmen of he echnology n hs feld. A Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna Elecrc rcu Theory The characerscs of he elecromagnec felds n he qauassaonary approxmaon ( D/ and / canno be dfferen from zero ogeher n he same olume: or D/ or / ) and he peculares of he maerals (elecrc conduce, sem-conduce and nsulang maerals) allowed he deelopmen of he elecrc crcu echnology. The elecrc crcus are used n many echnologcal applcaons for he reamen of nformaon sgnals and of power (elecrcal and elecronc deces, compuers, conrol deces, elecommuncaon sysems, elecrc power sysems). An elecrc crcu, made of nerconneced crcu elemens (ressors, nducors, capacors, dodes, ranssors operaonal amplfers), operaes followng he EM laws. Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 8

5 Inegral form of he EM equaons æ ö H dl ç ˆ ò ò J s Maxwell's law n ds l è S ø æ ö ˆ E dl- dφ nd Maxwell's law ç F ò ò n ds d l è S ø J ˆn ds- dq charge conseraon law d S D ˆn dsq Gauss's law S S J ˆn ds ˆn ds S Maeral laws H µ D ε E J σ E Three equaons are ndependen, he oher hree are dered from hem. Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 9 Elecrc rcu Theory dq ou - d q F L df e - d rcu elemen (crcu componen): s a regon of space nsde whch he elecromagnec phenomena are confned. J σ E ρ σ Inerconnecons: hey are conducng channels (usually cables) where he charges flows from a componen o anoher.. Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna

6 Elecrc rcu Theory The crcu model followng he lumped componen model approxmaon s based on he properes of maerals. Very mporan for hs s he elecrcal conducy dfference beween conducng and nsulang maerals (s ond /s Insul 5 ). As as resul, he crcu model s a ery good approxmaon. harged parcles moe nsde conducng maerals whch consue he elecrcal connecons beween he crcu elemens. These connecons are mmersed no nsulang maerals where charged parcles can no moe. Therefore he crcu elemens and he nerconnecons beween he consue a crcu physcally ery well defned ha can be modelled n a ery smple drec way. The elecrcal curren densy and s ube lnes are presen nsde he connecors and no ousde of hem n he nsulang maeral. Hence a conducng cable connecng he crcu elemens concde wh a flux ube of J. Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna Elecrc rcu Theory Assumpons The lumped crcu model assumes ha he elecromagnec (EM) phenomena ac only nsde crcu elemens and he mescales of he elecrcal quanes araons are much larger han he propagaon delays (he elocy of lgh s consdered o be nfne). Therefore he selfmposed consrans of he crcu model approxmaon are:. The change of he magnec flux n me ousde crcu elemens s zero. "# "$. The me araon of he charge nsde conducors s zero. "% "$ 3. The propagaon me neral of a sgnal from a crcu elemen o an oher one s equal o zero. Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna

7 Elecrc rcu Theory Assumpons. Assumpon allows o consder ha ousde he crcu elemens he elecrc feld s conserae. Hence he elecrcal enson (he olage) beween wo pons ousde crcu elemens s gen by an elecrc poenal dfference.. Assumpon allows o consder ha nsde connecors he dsplacemen curren densy J D D s equal o zero and ha he oal curren densy J JJ D s only gen by he conducon curren densy J r u. Hence nsde conducors he conducon curren densy s solenodal (,J ) and he curren s consan along he connecor. 3. Assumpon 3 allows o consder he sgnal propagaon me from a crcu elemen o an oher one hrough a connecor equal o zero. Therefore he sgnal (curren or olage) a he ex of a crcu elemen s equal o he sgnal reachng he crcu elemen o whch he frs one s conneced boh for me consan or me dependen currens and olages. Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 3 Elecrc rcu Theory The elecrcal curren densy and s ube lnes are presen nsde he conducng maerals of he connecors only and are equal o zero n he nsulang maerals ousde he conducors. Hence a conducng cable connecng he crcu elemens concde wh a flux ube of J. S Insde conducors and, herefore, nsde cables, he conducng curren densy J s solenodal. Hence he oal flux of J, whch s he oal S curren ou, flowng ousde of a segmen of a flux ube of J, made of he closed surface S S S L s equal o : n S (n) S (n) SL (n) where n s he unary ecor perpendcular o he closed surface, dreced ousde of. n ndcaes he curren drecon. The curren SL (n) s zero as he flux lnes of J are angenal o S L and no curren flows ou of S L. S S L n n n n S n Flux ube of J Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 3

8 Elecrc rcu Theory Hence follows ha S (n) S (n) When choosng he unary ecor n n he oppose drecon of n, s n - n and S (n ) - S (n). Therefore from he solenodaly of J resuls ha S (n ) S (n) Ths ald for any segmen of a flux ube of J. Therefore he curren hrough any cross secon of a flux ube remans consan. Ths means ha curren hrough any cross secon of a connecor or an elecrc cable s consan. S S L n n S n n S n S n Flux ube of J Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 5 Applcaons Elecrc rcu Theory An elecrc crcu s defned as he nerconnecon hrough elecrc conducors of crcu elemens (ressors, nducors, capacors, dodes, ranssors operaonal amplfers). The crcu heory ams o smulae and o predc he elecrcal behaor of he physcal crcus for he analyss and he desgn of hem (o enhance he performance, o decrease her cos, o analyze all workng condons, o sudy he faul condons, he hermal effecs, he endurance, ec.) ü dmensons: negraed crcus, h-f crcus, compuers, elecronc deces, elecommuncaon sysems, elecrcal power generaon, ransmsson and ulzaon sysems ( -3-6 m);; ü enson: -6 V (nose analyss deces) - 6 V(elecrcal power sysems);; ü curren: -5 A ( fa: elecromeers) - 6 A (power sysems);; ü frequency: (drec curren) - 9 Hz ( GHz: mcrowae crcus, compuers);; ü power: -4 W (rado sgnals from galaxes) - 9 W (elecrc power saons). 6 Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna

9 rcu Elemen The Two Termnal rcu Elemen Ø The wo ermnal crcu elemen s a crcu elemen conssng of a closed surface S from whch wo ermnals come ou. All EM phenomena are ace nsde S. Here E can be non-conserae. Ousde S all EM phenomena are slen, he E feld s conserae. S has a sheldng funcon. The crcu elemens are conneced o each oher hrough her ermnals. Ø The saus of he elemen s descrbed by curren and enson:,. Ø As a consequence of he charge conseraon low, he curren flowng hrough s he curren enerng from one ermnal and gong ou from he oher one. Ø Ousde S, E s conserae. Therefore ousde S a poenal funcon exss and s defned by he relaon E -. and are alues of a whch ermnal and ermnal are and s he correspondng poenal dfference (olage) beween hem. haracerscs of he Two Termnal Elemen rcu Elemen The wo ermnal elemen (dpole) s a crcu elemen wh wo ermnals hrough whch s conneced o oher crcu elemens. The ermnal are he nodes of he crcu. The crcu elemens are he branches of he crcu. > - > branch enson (or branch olage). s branch curren. In a branch s pose when eners no he ermnal a hgher poenal (ermnal n hs case) and goes ou from ermnal a lower poenal. The relaon beween and, gen by he elemen equaon f (), s characersc of each crcu elemen. The work made by he feld E on he charges whch flows hrough a branch cross secon per me un s he wo ermnal elemen elecrcal power : p() lm æ ö dq q d d ç ò D E l D d ò E l D è ø () () _ 8

10 n-termnal rcu Elemen Elecrc crcu elemen hang n ermnals wh n greaer han wo s sad n-ermnal crcu elemen. A reference ermnal s defned (M.). As a consequence of he charge conseraon: M. 3. n- As E s conserae ousde of he n-ermnal he followng defnons are made: M. - M. M. - M. 3 M..3 - M. n- M.n-. - M. 3 M. M. 3 n-ermnal M. n- n- Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna M. n- M. n- 9 n-termnal rcu Elemen A n-ermnal s descrbed by n- pars of alues, n- currens and n- olages. Hence s equalen o n- wo ermnal elemens wh a common ermnal.,, 3,, n-,, 3,., n- 3 3, 3,, n- n-, n- n-, n- n- Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna

11 n-termnal rcu Elemen A hree ermnal elemen corresponds o wo dpoles wh wo common ermnals. A four ermnal elemen correspond o hree dpoles wh hree common ermnals ec , 3 3, 3, 3 4 3, 3, 4, 4,,, Elecrc rcu In a crcu he nerconnecons among he crcu elemens are realzed hrough conduce cables (assumed as deal conducors, σ ). In he crcu heory he elemens are he branches of he crcu. The nerconnecons are he nodes of he crcu. A crcu s characerzed by he number n of nodes and he number r of branches. rcu wh nodes (n ) and 7 branches (r 7) Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna

12 Krchhoff s urren Law (KL) Krchhoff s Laws A closed surface S, whch passes hrough some nodes of he crcu bu no hrough crcu elemens, s consdered. No charge araon s assumed nsde conducors and J here s solenbodal. Hence for he charge conseraon law he oal currens enerng S s equal o zero: When S conans only one node, he followng corollary s obaned: n å k k The algebrac sum of he currens enerng a node s equal o zero S 4 5 Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna S 3 Krchhoff s Laws - Node Equaons For he nodes A,, and from he KL s: (A) () 3 5 () A ha s he resul of he addon of eq.s A, -, and. Ths s saed as follows: In a crcu n- node equaons are lnearly ndependen. Each equaon akes no accoun a leas a new curren whch does no appear n he oher equaons. In he equaon dered from he KL for node D only currens appear, whch 3 D are presen n he equaons of he 3 oher hree nodes. Ths equaon s a lnear combnaon of he ohers. For node D s: 4

13 Krchhoff s Tenson Law (KTL) The feld E s conserae n he regon ousde of he crcu elemens. Therefore he sum of he poenal dfferences on a closed pah l, whch connecs nodes and does no nersec crcu elemens, s equal o zero. A. HI IA Krchhoff s Laws G F E H I D A When l connecs he nodes of a crcu loop, he followng corollary s obaned. : The algebrac sum of he olages of he branches of a loop s equal o zero. m å k k l In an elecrc crcu a loop s defned as a closed pah passng only once hrough eery node n he pah. 5 Krchhoff s Laws - Loop Equaons The loops ADA and AA sre consdered. From he KTL follows (a) - 5 (b) A Each equaon akes no accoun a leas a new olage whch does no appear n he oher equaons. In he equaon D 3 dered from he KTL for loop ADA 3 only olages appear, whch are presen n he aboe equaons. Ths equaon s a lnear combnaon of he wo aboe menoned equaons. Ths s saed as follows: In a crcu r-n loop equaons are lnearly ndependen. Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 6

14 Node and Loop Equaons In a elecrc crcu conssng of dpoles, for each dpole a branch s defned. In he fgure r 5 and n 4. From he KL and he KTL follows: KL for each node: Σ n n KTL for each loop: Σ m m S A In he fgure r 5 and n 4: Ø n 3 lnearly ndependen node equaons, Ø r n lnearly ndependen loop equaons The equaons are of he followng ype: Ø for loop ADA: Ø for node A: D Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 3 l 7 Topology Equaons The saus of a crcu of r branches and n nodes s descrbed by r branch currens and r branch olages, whch are r quanes. Therefore r (r branch currens and r branch olages) are he unknown quanes of he analyss problem. From he opology of he crcu, he number of branches, he number of nodes, and her connecons, r lnearly ndependen opology equaons are obaned. n- are he node equaons dered from he KL. r-n are he loop equaons dered from he KTL. In order o hae an unque soluon of he analyss problem, oher r equaons are necessary. They are gen by he elemen equaons whch sae he 8 relaon beween curren and olage for each branch. Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna

15 Elemen Equaons Independen Volage Source Ideal essor s, s (ons.) - () () [Ohm s law] [Ω (Ohm)] elecrc ressance s - s - s rcu analyss: an example The problem of analyss The opology of he crcu and he ndependen olage sources ( s, s ) and he ressances (,, 3 ) are he problem npu. The branch olages and he branch currens (,, 3,,, 3 ) are he problem oupu.. Drec he branches (defne he pose drecon of he currens,, 3 ).. Defne he nodes and he loops for he ndependen opology equaons (n he crcu of he fgure: n-;; r-n). 3. Defne he drecon of each loop. 4. Wre he opology equaons (KL and KTL) and he elemen equaons: s s ( hese are 6 eq.s n 6 unknown wh an unque soluon) y subsung he elemen eq.s no he opology eq.s: s s s 3 3 s s A From whch he branch currens are obaned (,, 3 ). y subsung he currens,, 3 no he elemen eq.s he 3 branch olages,, 3 are dered.

16 Lumped rcu Approxmaons The lumped elecrc quanes n a crcu can hae rapd or slow me araons n comparson wh he rans mes whn he crcu. Ø The assumpon of lumped crcu s ha he rans mes whn he crcu are much less hen he me araon of he elecrc quanes consdered. When reang waes (elecrc quanes expressed by snusodal funcons n me), he lumped crcu assumpon consders hem propagang nsanly whn he crcu. Ø The rans me for he d propagaon of a sgnal from A o s A d/ d/c ( c 3 8 A m/s s he speed of lgh). For a snusodal sgnal wh a frequency f and perod T wh f /T, and wae lengh λ c/f, s T λ/c and A d/c. The assumpon of lumped crcu needs ha: A << T d << λ Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 3 Lumped rcu Approxmaons ü Elecrcal power: f 5/6 Hz λ 6/5 km ü Mcrowaes: f MHz λ 3 m ü ompuer clock: f 3 GHz λ cm The anenna of he fgure recees a sgnal of a frequency of MHz correspondng an angular frequency ω πf π 8. In A s A () V sn ω V sn(π 8 ) In he sgnal arres afer a me Δ d/c,5/3 8,5-8 s. Therefore n a he me he sgnal s () A (- Δ) V sn[π 8 ( - Δ)] V sn[π 8 ( -,5-8 )] V sn(π 8 - π) - V sn(π 8 ) - A () A d,5 m 3

17 rcu Elemens Two Termnal Elemen - Dpole Elemen Equaon The relaon beween he branch curren flowng hrough he elemen and he branch olage, whch s he poenal dfference beween he ermnals of he crcu elemen, defnes he behaor of ha elemen whn he crcu. Ths relaon s he elemen equaon (sad also he - characersc). urren conrolled elemen f() he curren s he ndependen arable. Volage conrolled elemen g() he olage s he ndependen arable. Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna rcu Elemens Two Termnal Elemens Elemen Equaon The elemen equaon, whch defnes he relaon beween he branch curren flowng hrough he elemen and he branch beween he ermnals of he crcu elemen, s deermned by he physcal phenomena caused by of he elemen. Passe Two Termnal Elemens In he passe elemen conenon he curren eners no he elemen from he pose ermnal. In he elemen, as n ressors, he charge s dsplaced from he hgher poenal o he lower poenal due o he pose poenal dfference. Therefore he energy resuls o be dsspaed. In passe elemens he energy s always pose or equal o zero. Passe dpole conenon w() p ò - - (' ) (') d' 34 ³

18 rcu Elemens Two Termnal Elemens Ace Two Termnal Elemens In he ace elemen he curren eners no he elemen from he negae ermnal. The curren flows from he negae o he pose ermnal. The crcu elemen s dong work n mong charge from a lower poenal o a hgher poenal. Elecrc power sources (ndependen enson sources and ndependen curren sources) are ace elemens. Ace dpole p - S (As saed by he passe elemen conenon he branch olage -V S ) Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 35 Elemen Equaons Lnear and non-lnear wo ermnal elemens q Lnear elemen: he elemen equaon consss of lnear operaors. d () a b () c d ò ()d d example: () q non-lnear elemen: he elemen equaon s non-lnear () a' b' example: () () Tme-ndependen and me-dependen elemens: q me-ndependen elemens: he elemen equaons do no depend on me (n eq.s and a, b, c, d, a and b are consan). q me-dependen elemens: he elemen equaons are medependen (n eq.s and a, b, c, a and b depend on me). Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 36

19 Elemen Equaons Dsspae and Sorage Elemens q Elemens whou memory: he elemen equaon expresses he relaon beween and a he same me (In hs case he passe elemens are dsspae). () a b () c sn [ () ] (non-lnear whou memory) q Elemens wh memory: he elemen equaon expresses he relaon beween and a dfferen mes. d () a b () c d (lnear dpole wh memory) () ò a (') d' (lnear dpole wh memory) - The elemens wh memory sore energy, whch can be rereed a a laer me. These elemens are also called sorage elemens. 37 Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna Ideal Elemenary Elemens essors, capacors, nducors, enson sources, and curren sources. The Ideal Elemenary Elemens are he deal ressor, he deal capacor and he deal nducor descrbed by he lnear expressons gen below. Each of hese deal elemenary wo ermnal elemens represens a sngle elemenary EM process. In a real elemen a sngle elemenary process s neer presen. Ideal ressor _ () () Ideal apacor q () - ò - (') d' Ideal nducor - d () L d

20 Ideal Tenson Source (Indpenden Tenson Source) Ideal urren Source (Independen urren Source) s - - s (os.) - s, s - s, onrolled Sources (Dependen Sources) - μ r Tenson conrolled enson source - r m r urren conrolled enson source g m r Tenson conrolled curren source α r urren conrolled curren source Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 39 Elemenary Ideal Dpoles The essor The ressor s a passe elemen whch dsspaes energy. f() - q Lnear me ndependen ressor (dsspae passe elemen): () () [Ohm s law] p() () () () - ressance (SI un ohm [Ω]), - G / conducance (SI un semens [S]), ϕ an φ Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 4

21 The Lnear essor The elecrcal ressance of a crcu elemen s he parameer ha quanfes s propery o oppose he curren: / The ressy ρ [Ωm] of a maeral quanfes s propery o oppose he flow of elecrcal charges: r s where σ [S/m] s he elecrcal conducy (J σ E) S cross secon l In a crcu elemen wh a consan cross secon S and a lengh l, where ρ s unform n he whole olume, he ressance of s expressed by: l r S l s S Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 4 Elecrcal essy In he fgure he ressy maerals usually ulzed n echnologcal applcaons s lsed. The common uses of hem as nsulang, semconducng, and conducng maerals, are repored oo. 4

22 The Ideal Volage Source Ideal Dpoles The deal olage source s an ace elemen. I keeps he enson s beween s ermnals ndependenly from he curren flowng hrough. s, s s - - s - The symbol a he rgh hand sde s used for D olages. Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 43 The eal Volage Source To smulae a real olage source a ressor n seres wh he deal source s consdered. s - When a arable load L conneced o a real source, from he KTL follows: - Þ L L - L L s L L s s s L L - s s s L L Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 44

23 Ideal Dpoles The Independen Ideal urren Source The ndependen deal curren source s an ace elemen. I keeps a curren s flowng hrough ndependenly from he olage. s, s - s Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 45 The eal urren Source To smulae a real curren source a ressor n parallel wh he deal source s consdered. s s s - / When a arable load L s conneced o a real source, from he KL follows : - and Þ s L L L L s s A - L - s L s L / L L Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 46

24 The apacor I consss of wo conducng plaes separaed by an nsulang maeral q q f() dq d Þ d d f() s he capacance [SI un: F (farad)]. I s gen by he rao beween he absolue alue of he charge on one of he capacor conducng plaes and he olage beween hem. When he nsulaor (delecrc) of a hckness d, placed beween he wo armaure s homogeneous, s: A e d Elemenary Ideal Dpoles For he lnear me ndependen capacor s: d() q() () Þ () d ε s he delecrc consan of he nsulang maeral. q - q Insulaor onducng plaes : A plae cross secon q - -q - - d 47 The lnear me ndependen capacor - q() (), q dq d () dq d d d () d d () ( ) e ò - Elemenary Ideal Dpoles (') (') d' q() ò (q') dq' q() ò d () d () (') d ', > q' dq' q [(- ) s assumed] q() - (') d ', Energy sored by he capacor - elecrosac energy - a he me : () [A - q s assumed o be zero]

25 The Lnear apacor Possble q, d, d e q q For a D curren a capacor s an open crcu (ressor wh ): Ø When he olage s consan n me, he curren s equal o zero ( d/d). Ø When a capacor s conneced o a baery becomes charged (q ). q The enson and he charge of a capacor canno ary nsananeously. Ø As saed by he capacor equaon q d/d and d/d, a olage dsconnuy mples an nfne charge and an nfne curren, ha s no physcally possble. Therefore he capacor opposes any sharp araon of he enson. Ø As saed by he capacor energy relaon, any nsananeous araon of he charge mples an nsananeous araon of he energy. Hence an nfne power s necessary o hae hs araon. Ths s no physcally possble. q The capacor s a passe elemen. I does no dsspae energy. The energy s sored n he form of elecrosac energy. The energy s used o creae an elecrc feld due o he deposon of he charges carred by he curren on boh conducng plaes. Ths energy s gen back when he curren changes s drecon. No Possble Elemenary Ideal Dpoles The Inducor I consss of wndngs around a core of ferromagnec maeral. The curren flows hrough he wndngs and generaes a magnec flux. A me araon of hs flux nduces a olage. - F f () Φ df d Þ f () d d N A L µ l μ s he magnec permeably [SI un: H/m]. q For he lnear me ndependen nducor s: d F () L () Þ L N - number d of urns L s he nducance [SI un: H (henry)]. I s gen by he rao beween he magnec flux generaed by he curren and he curren. A: ferromagnec core cross secon l: core lengh Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 5

26 The lnear me ndependen capacor - F () L () Φ df d () L d d () L () ( Elemenary Ideal Dpoles d d ) L ò d L (') d', () d " > () L ò - (') d', [(- ) s assumed] Energy sored by he nducor elecromagnec energy - a he me : e L ò (') (') d' ò L ' d' L - [(- ) s assumed] Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna The Lnear Inducor F d L, L, e d L L q For a D curren an nducor s a closed crcu (essor wh ): Ø When he curren s consan n me, he olage s equal o zero ( L d/d). q The curren n an nducor canno ary nsananeously: Ø As saed by he nducor equaon L d/d, a curren dsconnuy mples an nfne enson, ha s no physcally possble. Therefore he nducor opposes any sharp araon of he curren. Ø As saed by he nducor energy relaon, any nsananeous araon of he curren mples an nsananeous araon of he energy. Hence an nfne power s necessary o hae hs araon. Ths s no physcally possble. Possble No Possble q The nducor s a passe elemen. I does no dsspae energy. The energy s sored n he form of magnec energy. The energy s used for he creaon of he magnec feld due o he curren flowng n he wndngs. Ths energy s gen back when he curren changes s drecon. 5

27 Two Termnal Elemens Open rcu I can be consdered as eher of he followng elemens: - a curren source wh s - a ressor wh Elemen eq.:, losed rcu I can be consdered as eher of he followng elemens: - a olage source wh s - a ressor wh Elemen eq.:, s s ca cc Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 53 Ideal Dode Two Termnal Elemens per < per > pn-juncon Dode The operaon regon s for > A. For < A he dode burns ou. é æ ö ù I ê ç ç - s exp ú ë è VT ø û I s ( μa) sauraon curren V T kt/e (.6 V) hermal enson Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna A I s 54

28 Seres essors Two or more wo ermnal crcu elemens are sad o be n seres f he curren from one elemen exclusely flows no he nex elemen. Therefore he same curren flows hrough each elemens one afer anoher n-,n n- n-,n k eq k k n- 3 n- n G eq n- k eq n n- G k - - k, G eq eq k G k ( ) Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 55,n,n...,n æ ç ç è Parallel essors Two or more crcu elemens are sad o be n parallel f he elemens share he same wo ermnals. Therefore he elemens wll hae he same olage. eq,n n...,n n... eq eq n å k å ö ø K k k,n eq eq n - - ( G, G ) G G k n n n k eq eq Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 56

29 essor n seres wh an open crcu essor n seres wh a closed crcu eq n n ressors n seres wh equal eq /n essor n parallel wh an open crcu essor n parallel wh a closed crcu n ressors n parallel wh equal Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 57 Analys of a crcu Known quanes: Ω, Ω, 3 3 Ω, 4 Ω, 5 Ω, 6.5 Ω, 7 Ω, 8 6 Ω, 9 Ω, Ω. V s 4 V. Deermne he branch currens. eq eq eq 3.33 V 4 eq / 4 ; 5 eq / A 5.67 A eq eq W V s eq V s eq / A eq eq 3 8 eq 3 / eq eq 3W eq 3 8 V s - eq A Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna

30 6-7 4eq A 4eq 3eq 6-7 3eq 4eq eq - V 3eq 3eq eq eq.5 W 3 3 3eq V s 5eq 4eq 8 - V 4eq 4eq / 4eq ; 8 8 / 8 4eq A; A 4eq 4eq 6 3W 7 3eq 4eq 4eq V s 9 5eq -V s 5eq V s 5eq - V 9 -V s / Eq - A 5eq 5eq Eq Eq 4eq 4eq W 9 4 W 8 8 5eq 5eq Eq 9 Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna V s V s Tenson Dder eq s s eq Vs VS From he KTL s - V Þ A Vs Vs s V s Vs - Vs - V - V - - urren Dder eq eq Is From he KL s - I Þ s I s A Is - Is - Is - Is - Is Is I S 6

31 Wye and Dela essor onnecons A sysem of hree ressances can be dela conneced or wye conneced. I can be more conenen o work wh a wye nework n a place where he crcu conans a dela confguraon. A wye nework a dela nework can operae n an equalen way. Dela onnecons Wye onnecons Δ Δ3 Y Y 3 Δ 3 Y3 Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna Wye and Dela essor onnecons eween node and node, f node 3 s no conneced, n he wye and he dela connecon here are he followng ressances ( Y) Y Y Δ ( ) Δ //( Δ Δ3 ) If node 3 s no conneced he same curren has o correspond o he same olage. Ths as o hold for he brances - 3 and - 3 when node and areno conneced. Therefore : ( Δ) ( Υ), 3 ( Δ) 3 ( Υ), 3 ( Δ) 3 ( Υ) Y Δ Y Y Y Y3 Y3 Y Y3 Δ Δ Δ3 Δ Δ Δ3 ; Y Δ Δ3 3 Δ Δ ; Δ Δ Y3 c Δ3 Y Δ Δ3 Δ Δ Δ3 Y Y3 3 ; Δ Y Y Y Y3 Y3 Y ; Δ3 Y Y Y Y3 Y3 Y Y Y

32 Y-Δ Transformaon Each ressor of he wye connecon s he produc of he wo ressors of he dela connecon conneced o he same node, dded by he sum of he hree ressors of he dela connecon. Δ3 Y Δ Y Y3 Δ Y Δ Δ3 Δ Δ Δ3 ; Y Δ Δ ; Δ Δ Y3 c Δ3 Δ Δ3 Δ Δ Δ3 3 Each ressor of he dela connecon s he sum of all he producs of he ressors of he wye connecon wo by wo, dded by he ressance n he oppose branch of he wye connecon. Δ Y Y Y Y3 Y3 Y Y3 ; Δ Y Y Y Y3 Y3 Y ; Δ3 Y Y Y Y3 Y3 Y Y Y Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 63 Y-Δ Transformaon Known quanes: 3 Ω, 3 Ω, 3 3 Ω, 4 Ω, A 3 Deermne he equalen ressance beween he nodes and. 4 AΔ 3 3 Δ 3 3 Δ Ω 9 Ω 9 Ω A Δ Δ AΔ 4 Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 64

33 Y-Δ Transformaon Known quanes: 3 Ω, 3 Ω, 3 3 Ω, 4 Ω, AΔ Δ Δ 9 Ω eq 4 AΔ Δ 4 AΔ AΔ Δ 4 Δ,385 Ω A Δ Δ AΔ 4 Δ A AΔ 4 eq Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 65 Y-Δ Transformaon Known quanes: Ω, Ω, 3 Ω, 4 3 Ω, 5 Ω, 6.5 Ω, V s 4 V. Deermne he power delered by he olage generaor. AΔ Δ Δ Ω 6 Ω 6 Ω A 4 V s 3 - Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 5 6 A 4 V Δ s - Δ A 6 Δ 66

34 Y-Δ Transformaon 4 V s Δ - 6A 4 4 Δ Ω 4 Δ A Known quanes: V Δ s Δ Ω, Ω, 3 Ω, - A 4 3 Ω, 5 Ω, 6.5 Ω, Δ V s 4 V. 6 AΔ Δ Δ 6 Ω A A A 6A 6 AΔ,46 Ω 6 AΔ V s - Δ A A 4 6A,46 Ω eq Δ A,745 Ω A Δ V s V s - 4 eq ( eq ) 4,57 A p V s 58,88 W 67 apacors n seres doe: k... k (')d' ò n- (')d' k ( ) n- - n n- (')d'... ò ò ò n- n- (')d' ( ) ( )... ( n- ) eq - n æ ç ç è... n- ö ò (')d' ø ( ) ( )... ( n- ) ò eq (')d' ( ) doe : eq... Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna n- 68

35 apacors n parallel... n doe : k k d d d d d d... d n d ( ) d... n eq d d d doe : eq... n n n n eq n Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 69 Indcors n seres L L L n n- n n-... n- doe : k L k d d d L d L d d... L n- ( ) d L L... L n- L eq d d d d d doe : L eq L L... L n- L eq - n Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 7

36 Inducors n parallel L L n... n doe : k (')d' L k ( ) k L (')d' L (')d'... L n (')d' ( ) ( )... n ( )... L L L (')d' n ( ) ( )... n ( ) (')d' ( L ) eq doe : L n... L eq L L L n n L eq n Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 7 Magnec Flux n Inducors The Magnec flux Φ, lnked wh he wndng, hrough whch he curren flows, s generaed by ( Maxwell s law or Ampere s law): F f () N S For a lnear nducor s Φ F L For he Maxwell s law (also Faraday s law or nducance law) s df d d L d Ferromagnec core Number of urns Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 7

37 Lnear Tme Independen oupled Inducors The magnec flux lnked wh he crcu s generaed by wo currens: he flux componen generaed by and he flux componen generaed by : Φ Φ ( ) Φ ( ) For he lnear case s Φ L M Here L and M are he self nducance and he muual nducance respecely. For me ndependen crcu elemens resuls o be d M F d d L d d d M depends on each of he wo crcu and on her relae poson. M can be pose or negae Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 73 Lnear Tme Independen oupled Inducors q The wo crcus, wh or whou conacs beween hem, affec each oher by means of he magnec feld generaed by he currens flowng hrough hem. They are sad o be magnecally coupled. For wo magnecally coupled nducors s Φ L M Φ L M I can be demonsraed ha M M. Hence resuls: d d L M d d d d M L d d M L L Spo conenon: M > when he spos correspond o ermnals wh currens flowng nsde boh of hem or ermnals wh currens flowng ousde boh of hem. Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna

38 Lnear Tme Independen oupled Inducors Spo onenon When he spos correspond o he wo ermnals wh currens flowng nsde boh of hem or ermnals wh currens flowng ousde boh of hem, he muual nducance M s assumed o be pose: M > M > M < M < Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 75 Lnear Tme Independen oupled Inducors Magnec Energy Sored n oupled Inducors The magnec energy sore n he magnecally coupled nducors n he me neral from - o, assumng ha (- ) (- ), () I and ()I, s gen by e ( -, ) ò [ ] e - - ò ò - ì d í L M î d [ M ½ L ] d ½ L d ( -, ) ½ L I M I I ½ L I é d ê ë d L L d ù L d ú û M d ü ý d d þ

39 N-Por Elemens q In p-ermnal crcu elemen wh p een, he ermnals can be organzed n pars. When he curren flowng nsde he frs ermnal of each par s equal o he curren flowng ousde he second ermnal of, he elemen s an N-por elemen wh N p/.. q The quanes defnng a N-por elemen are:. N ;. N p()... N N T ε p(') d' When ε N N - q The N-por elemen equaon s gen by N scalar equaons relang k o k. (k,,3,,n). 77 Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna N... N he N-por elemen s passe. Two Por Elemens HI-FI deces Two wndng ansformer - Inpu por: por - Oupu por: por - Por enson:, - Por curren:, (, ) - Lnear wo por elemen é ê ë ù ; ú û A c p T é ù ê ë ú û a a b b c a a b b c p Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 78

40 ace elemen branch branch curren branch enson capacor capacance crcu componen crcu elemen closed crcu conducance conducng plaes, armaures conducy conrolled source curren conrolled elemen elemeno ao ramo, lao correne d ramo ensone d ramo condensaore capacà componene crcuale elemeno crcuale crcuo chuso conduanza armaure d condensaore onducblà Termnology Englsh Ialan generaore conrollao elemeno conrollao n correne coupled nducors dela connecon delecrc consan delecrc, nsulaor dode dsspae elemen dsrbued parameers elecrc crcu elecrc power elecrc ressance elecromagnec energy elecrosac energy elemen equaon nduor accoppa connessone a rangolo cosane delerca delerco, solane dodo elemeno dsspao paramer dsrbu crcuo elerco poenza elerca ressenza elerca energa eleromagneca energa elerosaca equazone caraersca, equazone cosua ndependen curren generaore d curren curren dder parore d correne source ndpendene 79 Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna Termnology Englsh Ialan ndependen olage source nducance nducor lnear (non- lnear) elemen lumped parameers Krchhoff s curren law Krchhoff s enson law monolhc chp muual nducance n- ermnal elemen n- por elemen wo- por elemen node generaore d ensone ndpendene nduanza nduore elemeno lneare (non lneare) paramer concenra legge d Krchhoff delle corren legge d Krchhoff delle enson chp monolco muua duanza eemeno ad n pol elemeno a n pore elemeno a duepore nodo, polo open crcu parallel connecon passe elemen pn- juncon ressor ressance ressy self nducance seres connecon sorage elemen enson dder wo ermnal crcu elemen olage conrolled elemen wyeconnecon crcuo apero connessone n parallelo elemeno passo gunzone pn ressore ressenza ressà auonduanza connessone n sere elemeno con memora parore d ensone elemeno crcuale a due ermnal, bopolo elemeno conrollao n ensone connessone a sella Deparmen of Elecrcal, Elecronc, and Informaon Engneerng (DEI) - Unersy of ologna 8