3. Quasi-Stationary Electrodynamics

Transcription

1 3. Qus-ttonry Electrodynmcs J B 1 Condtons for the Qus- ttonry Electrodynmcs The Qus-ttonry Electrodynmcs s chrcterzed y 1 st order tme ntercton etween electrc nd mgnetc felds. In qus-sttonry EM, n the sme pont the tme dervtves of D nd B cn not e dfferent from zero smultneously. Posson s equtons ecome: A = - µj y V = - % & Ther solutons re: V P(t) =,-& / 0 () A P(t) = 5,- J 0 () where r nd J re the sources. d d z d P 0 r P x

2 Condtons for the Qus-ttonry EM Therefore, n the Qus-ttonry Electrodynmcs pproxmton the vector potentl nd sclr one n pont P t the tme t depend only on the condtons of the EM sources of the electrc nd mgnetc felds t the sme tme, ndependently from the dstnce etween the plce t whch the sources re plced nd P. Ths mens tht the tme needed from the EM sgnls to trvel from ther sources (n P 0 ) to the pont where re consdered (n P), nmed trnsent tme, s neglected. r/u 0 Where u = (e µ) -0.5 = c(e r µ r ) -0.5 s velocty of the EM sgnls n the medum nd r s the dstnce etween P nd P 0. If the sptl dstrutons of r nd J re known t ny gven tme, the expressons of V nd A re determned t ll tmes s f the potentls were only functon of the poston nd not of tme. The EM dynmcs of the system s pproxmted y sequence of sttonry sttes. 3 Condtons for the Qus-ttonry EM Ø The ssumpton of quy-sttonry EM n lumped crcuts s tht the trnst tmes wthn the crcut re much less then the tme vrton of the electrc qunttes consdered. When EM qunttes expressed y snusodl functons n tme, the qus-sttonry EM ssumpton consders them propgtng nstntneously wthn the crcut. Ø The trnst tme for the propgton of sgnl r from A to B s t AB = r/u = = r/c (u = c = m/s s A B the speed of lght). For snusodl sgnl wth frequency f nd perod T wth f = 1/T, nd wve length λ = c/f, t s T = λ/c nd t AB = r/c. The ssumpton of qus-sttonry EM n lumped crcuts needs tht: t AB T r λ 4

3 Condtons for the Qus ttonry EM Electrcl power: f = 50/60 Hz λ = 6000/5000 km Mcrowves: f = 100 MHz λ = 3 m Computer clock: f = 3 GHz λ = 10 cm The ntenn of the fgure receves sgnl of frequency of 100 MHz correspondng n ngulr frequency ω = πf = π At the tme t the potentl n A s: v A (t) = V 0 sn ωt = V 0 sn(π 10 8 t) In B the sgnl rrves fter tme Δt = r/c = 1,5/ = 0, s. Therefore n B t the tme t the sgnl s v B (t) = v A (t- Δt) = = V 0 sn[π 10 8 (t - Δt)] = = V 0 sn[π 10 8 (t - 0, )] = = V 0 sn(π 10 8 t - π) = = - V 0 sn(π 10 8 t) = - v A (t) A B r = 1,5 m 5 Lumped Crcut Approxmton The ssumpton of quy-sttonry EM llows the lumped crcut pproxmton. The crcut elements re mde n such wy tht: B - 0 only nsde cols, t D - 0 only nsde cpctors, t B D - = = 0 n ll other regons t t The followng ssumptons re lso done: B t 0 - E 0 only nsde tenson genertors, - J = 0 n the delectrc etween the cpctor pltes, - 1/s 0 nsde resstors, - 1/s = 0 (s = ) nsde conductors. L?D? 0 C R - c d 6 e s

4 Poyntng s theorem expresson s: 4 J dτ tht s: Lumped Crcut Approxmton = DE 4 F d 4 d P g = P J P E P M P 4 B d d In regon the energy delvered y the mpressed electrc feld P g s dsspted y the Joule hetng P J, s used to crete the electrc nd the mgnetc felds (P E nd P M respectvely), nd s flowng outsde through ts externl surfce y mens of the flux of the Poyntng s vector. D t In crcut where E s consdered to e dfferent from zero only nsde tenson genertors, D/ t 0 nd B/ t 0 respectvely nsde cpctors nd nsde cols (nductors) only, 1/s 0 nsde resstors nd 1/s = 0 nsde conductors the lumped crcut pproxmton s mde where nsde regons where only one of the term P g (tenson genertor power), P J (resstor power), P E (cpctor power), nd P M (nductors), s dfferent from zero. These regons re connected y conductors. P s dfferent from zero only when the externl surfce of s crossed y conductors. Lumped Crcut Approxmton As n crcut the electrcl conductvty of the conductor s much hgher thn tht of the surroundng regon ( 10 0 ), nsde conductors J 0 nd D t=0 (J t = J), nd nsde the delectrc of cpctors s D t 0 nd J=0 (J t = D t) the crcut consstng of the conductng cle nd the delectrc of the cpctor s flux tue of the totl current densty J t. B t 0 L?D? 0 C R - c d e s Therefore, s J t s solenodl vector J t = 0), the flux of J t through every cross secton of the flux tue s constnt wth: D d d = R R T t where c s the cross secton of the conductor nd d s the cross secton of the delectrc nsde the cpctor. 8

5 Lumped Crcut Approxmton Inductor - Insde col t s B t 0, ( E 0), D t = 0, E 0, 1/s = 0. Outsde the col t s B t = 0, E =0, E = - V. Hence n closed lne l from the node to the node nd from to gn wthout ntersectng the col nd the crculr surfce nsde t, t s: B t 0 dl = V Z -V V -V = V -V = 0 The col s wrpped round lner mterl of permelty µ. A closed surfce s tken round, volume contnng the col. s untry vector perpendculr to n ny pont of t. The EM power flowng nto through the termnls nd (Poyntng s theorem) s: - E x B d H d 4 t l L?D? 0 L C R 9 - c d e s Lumped Crcut Approxmton - E x d = 4 H d == P M where W M s the B-feld energy n. On the surfce t s B t=0 nd E = - V. Therefore t s: E x H = - V x H = - (VH) V H L - E x d = d - from dvergence (VH) = 0 - E x d = - = V d R \ V d = - V d R ] V d on : : H = J = (V V ) = V where nd re the cross sectons of the ntersecton of the surfce wth the conductors ner nd. V nd V re the potentls n nd.

6 Lumped Crcut Approxmton The expresson of the the energy gong through nto, necessry to the ncrese of EM feld wthn the col, s: As - E x E = -?B? d = T^_ nd dl Z = - T`C Thus n terms of lumped prmeters the energy gong nto ecomes: T^_ = V = TF C W M = = B d 4 E5 = P M, where V = T`C F C s the lnked flux, lnked wth the col (flux through surfce wth closed lne l s ts order). Usully for col, F C s the flux through turn of the col tme the numer N of turns n the col (ths defnton of F C ssumes tht the flux through turn s the sme for ll turns of the col). L Lumped Crcut Approxmton Inductnce In order to express the B-feld energy y mens of lumped prmeters for lner mterl (B=µH) the nductnce L s defned s follows: W M = E L V = L Tc T^_ = L Tc V = L Tc The nductnce s defned s rto etween twce of the B-feld energy nd the squre of the col current. An lterntve defnton s tht L s the voltge nduced etween the termnls - over the current vrton per unt of tme. From the defnton of the B-feld energy tme dervtve t s: V = TF C = L Tc F C = L nd W M = E F C The thrd defnton of L s tht L s the B flux ntensty per unt of current. The nductnce L depends on µ of the col mterl nd the geometry of t. L

7 Lumped Crcut Approxmton Two Wndng Col elf nd Mutul Inductnce When n the col there re two wndngs wth two seprte currents, ech wndng s lnked wth the mgnetc flux nd ech current contrutes to the mgnetc flux. 1 The mgnetc flux F C1 lnked wth crcut 1 nd the flu F C lnked wth crcut re: F C1 = F 1 ( 1 ) F 1 ( ) Φ C = F ( ) F ( 1 ) where F 1 ( 1 ), F 1 ( ), F ( ), nd F ( 1 ) re respectvely the flux lnked wth crcut 1 nd generted only y 1, the flux lnked wth crcut 1 nd generted only y, the flux lnked wth crcut nd generted only y, nd the flux lnked wth crcut nd generted only y 1. L c d 13 Lumped Crcut Approxmton elf nd Mutul Inductnce The B-feld power flowng nto the col through the termnls - s due to F C1 nd tht flowng nto the col through c-d s due to F C : 1 L c d V 1 = T W EM(F C1 ) = T [W Tc EM(F 1 ( 1 )) W EM (F 1 ( ))] = L 1 1 M Tc 1 E 1 V cd = T W EM(F C ) = T [W Tc EM(F ( )) W EM (F ( 1 ))] = L E M Tc 1 These re the defntons of the self nductnces L 1 nd L, nd the mutul nductnces M 1 nd M 1. It cn e seen tht M 1 = M 1. Thus the tensons V nd V cd re: Tc V = L 1 M Tc E 1 ; V Tc cd = L E M 1 The totl EM power enterng the col through the crcuts 1 nd s: P M = T^_ Tc = V 1 V cd = T ( E L 1 1 M 1 1 E L ) 14

8 Lumped Crcut Approxmton Cpctor Insde of t t s D t 0 nd B t=0 ( E = 0 nd E = - V), nd J = 0. The EM power flowng nto s: - E x d = 4? d = = T e E 4 d = T^e = P E E V = T^e Chrges stored n Cpctor where W E = e E 4 d E A volume wth lterl surfce L of flux tue of D contnng d, totl cpctor delectrc regon, nd two surfces 1 nd nsde the surfce of ech of the two cpctor pltes. The closed surfce contnng ths volume s: = L 1. From the Guss theorem t s: d R = Q 1 Q where Q 1 nd Q re the chrges contned n ech plte of the cpctor. C 15 Lumped Crcut Approxmton Chrges stored n Cpctor R d However t s: d R = d R f d R f d R E = d R d = d R E = 0 Ths s s D s tngent to the lterl surfce of the flux tue, hence on L t s = 0. On 1 nd, whch re nsde the pltes of the cpctor, t s J = 0, thus J = s E = s D/e = 0 D = 0. Hence the ntegrls on 1 nd re equl to zero t: R d = Q 1 Q = 0 Q 1 = - Q = Q Ths s very mportnt result tht ndctes tht the chrges stored on the two pltes of cpctor re equl n module nd wth opposte sgn. 16

9 Lumped Crcut Approxmton Therefore the Poyntng s theorem expresson for feld flowng nto the volume d etween the cpctor pltes, gven y: - E x d = T^e = P E n terms of lumped prmeters ecomes: V = T^e Cpcty To express the EM energy stored n the cpctor y mens of lumped prmeters for lner delectrc (D=e E) the cpcty C s defned s: W E = E CV V = C V TV T^e TV = C V = C TV d The cpcty s defned s rto etween twce of the EM energy nd the squre of the tenson from to. An lterntve defnton s tht L s the current flowng from to over the tenson vrton per unt of tme. 17 Lumped Crcut Approxmton The chrge flowng through the current nto the cpctor plte s: Q(t) = (t j p ) dt = C \ dv kl c 1 Q(t) = C V c Ths s the thrd defnton of C whch s the chrge contned nto plte per unt of tenson. The cpcty C depends on e of the delectrc nd the geometry of t. From the defnton of C, t s: = C TV V = (tj ) dt = q 1 kl (tj ) dt 1 where: Q 0 = (t j ) dt 1 kl d W E = e E 4 d = CV E E dv = (t)dt 18

10 Lumped Crcut Approxmton Resstor Insde of t t s: D t=0 nd B t =0 ( E = 0 nd E = - V), nd 1/s 0. The EM power flowng nto s: - E x d = d = P 4 J V = Resstnce 4 DE F V = P J = s DE F d = c E FR E 4 ce FR E d C dl = dl = R s FR From these results the resstnce s defned s: R = P J = dl s FR Where l s the length trvelled y nsde the resstor nd C s the cross secton of the resstor perpendculr to the drecton of the resstor. The resstnce R depends on the electrcl conductvty nd the geometry. The current densty s ssumed to flow nto thredlke conductor of length l nd cross secton c. Resstnce Lumped Crcut Approxmton V = R V = R Alterntvely t s lso: 1 R = P J V Hence the resstnce depends on the electrcl conductvty nd the resstor geometry. The resstvty s defned s the rto etween the electrcl power enterng the closed surfce dsspted y Joule hetng, nd the squre of the current enterng termnl. The resstnce s lso the tenson per unt of current or the squre of the tenson over joule power. R s constnt for conductor n sttonry regme (dc regme). In qussttonry regme the skn effect s present nd R s ffected y t. The skn effect grows wth the ncrese of the frequency. 0

11 Lumped Crcut Approxmton Tenson Genertor Insde of t t s: D t=0 nd B t=0 ( E = 0 nd E = - V), E 0 nd 1/s = 0. The EM power flowng nto s: - E x d = - 4 J dτ = P g When the current densty s ssumed to flow nto the thredlke conductor of length l nd cross secton c nto the genertor, t s: - e s J d = c c dl = dl where dl s dfferentl vector long thredlke conductor nto the genertor. V = 4 J dτ = E s V = e s where e s = E dl nd the power flowng nto : V = e s V = e s s the defnton of tenson genertor or voltge source. 1 Lumped Crcut Approxmton Idel Crcut Approxmton It s possle to relze crcuts wth D t 0 only nsde cpctors nd D t 0 only nsde nductors. Thus qus-sttonry regme s relzed nd crcut pproxmton descred y lumped qunttes cn e done. Wth the ssumpton of del crcut pproxmton, the followng ssumpton re done: E 0 only nsde tenson genertors, J=0 nsde the delectrc of cpctors, 1/s 0 only nsde resstors, 1/s = 0 nsde conductors. The defnton of R, C, L, M, nd e g re del crcut elements n the lumped crcut pproxmton. They ssume tht there s lner relton etween nd V nd tht these constnt don t depend on tme. Inductors nd cpctors re memory elements, resstor s not.

12 Lumped Crcut Approxmton Element Equton The relton etween the rnch current flowng through the element nd the rnch voltge v, whch s the potentl dfference etween the termnls of the crcut element, defnes the ehvor of tht element wthn the crcut. Ths relton s the element equton (sd lso the -v chrcterstc). v 1 Current controlled element v = f() the current s the ndependent vrle. Voltge controlled element = g(v) the voltge s the ndependent vrle. Lumped Crcut Approxmton Element Equton The element equton, whch defnes the relton etween the rnch current flowng through the element nd the rnch v etween the termnls of the crcut element, s determned y the physcl phenomen cused y of the element. Pssve T1wo Termnl Elements In the pssve element conventon the current enters nto the element from the postve termnl. In the element, s n resstors, the chrge s dsplced from the hgher potentl to the lower potentl due to the postve potentl dfference. Therefore the energy results to e dsspted. In pssve elements the energy s lwys postve or equl to zero. Pssve dpole conventon w(t) = p t ò - v - v(t' ) (t') dt' 4 ³ 0

13 Lumped Crcut Approxmton Actve Two Termnl Elements In the ctve element the current enters nto the element from the negtve termnl. The current flows from the negtve to the postve termnl. The crcut element s dong work n movng chrge from lower potentl to hgher potentl. Electrc power sources (ndependent tenson sources nd ndependent current sources) re ctve elements. Actve dpole p - v g (As stted y the pssve element conventon the rnch voltge v = -V g ) 5 Lumped Crcut Approxmton Lner nd non-lner two termnl elements q Lner element: the element equton conssts of lner opertors. d t v(t) = (t) c d ò (t)dt dt t 0 exmple: (1) q Non-lner element: the element equton s non-lner v(t) = ' ' exmple: () (t) Tme-ndependent nd tme-dependent elements: q tme-ndependent elements: the element equtons do not depend on tme (n eq.s 1 nd,, c, d, nd re constnt). q tme-dependent elements: the element equtons re tmedependent (n eq.s 1 nd,, c, nd depend on tme). 6

14 Lumped Crcut Approxmton torge Elements q Elements wthout memory: the element equton expresses the relton etween nd v t the sme tme t (In ths cse the pssve elements re dssptve). v(t) = (t) c sn [ (t) ] (non-lner wthout memory) q Elements wth memory: the element equton expresses the relton etween nd v t dfferent tmes. d v(t) = (t) c dt (lner dpole wth memory) t v(t) = ò (t') dt' (lner dpole wth memory) - The elements wth memory store energy, whch cn e retreved t lter tme. These elements re lso clled storge elements. 7 Lumped Crcut Approxmton Idel Crcut Approxmton In ths pproxmton ctve elements (they produce EM energy) nd pssve (they sor or do not produce EM energy) crcut elements. Pssve elements: Idel resstor _ v Idel Cpctor q - v Idel nductor - v Idel coupled Inductor 1 - v v(t) = R (t) kl v(t) = (tj ) dt v(t) = L Tc v(t) = L Tc M Tc 8

15 Lumped Crcut Approxmton Actve elements: Independent ources Dependent ources Independent tenson genertor v = v s, v s - - Independent current genertor = s, v Tenson controlled tenson source v = μ v r - Tenson controlled current source Current controlled tenson source v = rm r - Current controlled current source = α r = g m v r s - 9 Lumped Crcut Approxmton Idel Crcut Approxmton An del crcut consstng of the crcut elements n the prevous sldes, s descred y the scheme n the fgure nd y the relton etween current nd voltge gven elow. v (t) r v r (t) r V 0 R r s - t d r ds 1 = Lr Mrs dt dt C ò r - M rs L r C r (t')dt' R (t) r r - V 0 Ths scheme refers to crcut rnch. The rnch current r enters y the postve rnch termnl () when the rnch voltge v r s consdered. 30

16 Lumped Crcut Approxmton In electrc crcut consstng of crcut elements wth two termnls (dpoles), ech dpole s rnch connected to the termnls of the other rnches. The connecton re the nodes of the crcut. In the crcut of the fgure there re fve rnches (r = 5) nd four nodes (n = 4). For closed lne l C tht doesn t ntersect crcut elements t s E = 0 nd E = - V. dl Z v = 0 z = 0 for ech loop (TL) v m For closed surfce C tht tht doesn t ntersect crcut elements t J = 0. v d = 0 n = 0 for ech node (CL) C d 3 l C 31 c Lumped Crcut Approxmton The krchhoff s lws (krchhoff s tenson lw - TL Gustv rchoff ) for the ssumptons mde re consequence of the mxwell nd lw (mxwell s 1 st nd nd lws Jmes Clerk Mxwell ) nd the chrge conservton lw (J solenodl on C ). If r s the numer of rnches of the crcut nd n s the numer of nodes, n crcut there re: Ø n - 1 lner homog. Indep. node eq.s (CL);; Ø r n 1 lner homog. Indep. loop eq.s (TL). In order to know the complete sttus of the crcut r rnch tensons nd r rnch currents ( r unknowns) hve to e determned. Ø r eq.s re the lner non-homog. rnch eq.s. A lner non-homogeneous set of r equtons n r un knowns s otned. C d 3 z v m = 0 n = 0 v r = f r ( r ) l C 3 c

17 kn Effect Qus-ttonry Electrodynmcs It s the tendency of n lterntng electrc current to ecome dstruted wthn conductor such tht the current densty s lrgest ner the surfce of the conductor, nd decreses gong nsde the conductor. The electrc current flows mnly t the "skn" of the conductor, etween the outer surfce t the cle rdus R nd level clled the skn depth d. 3.0 The skn depth s mesure of the depth.5 t whch the current densty flls to 1/e (0.3679) of ts vlue ner the surfce..0 The skn effect cuses the effectve resstnce R AC n the AC regme to ncrese wth the ncrese of the frequency n comprson to ts DC vlue R DC. R AC /R DC R/d 33 Qus-ttonry Electrodynmcs kn Effect Motvton The skn effect s ctve lso when consderng the qus-sttonry ssumpton. An lterntng current n conductor produces n lterntng mgnetc feld nsde nd round the conductor. When the ntensty of current chnges n tme, the mgnetc feld lso chnges ( B t 0). Ths cretes n electrc feld whch opposes the chnge n current ntensty (counter-electromotve force or ck EMF). The ck EMF s strongest t the centre of the conductor, nd forces the conductng electrons to the outsde of the conductor. An lterntng current my lso e nduced n conductor due to n lterntng mgnetc feld ccordng to the lw of nducton. An electromgnetc wve mpngng on conductor wll therefore generlly produce such current;; ths explns the reflecton of electromgnetc wves from metls. 34

18 Qus-ttonry Electrodynmcs kn Effect Motvton A cle s consttuted of two coxl cylndrcl tues of dfferent rdus nd of resstnces R 1 nd R (see fgure). In the nner nd outer conductor the currents 1 nd re flowng. They re produced y the snusodl voltge source v(t). v(t) ~ - 1 The lnes of flux of the mgnetc feld re crculr nd coxl wth the cylndrcl conductors. The totl flux s mde of two components. The frst component s externl to the two conductors F ex nd s generted oth y 1 nd y. The second component s generted only oth y 1, nd s n regon etween the two conductor F n. F n F ex B 35 Qus-ttonry Electrodynmcs kn Effect Motvton The geometry of the lnes of flux of F ex wth oth currents nd 1, nd ts lnkng wth them do not depend on whch of them generte the flux F ex. Hence t s: F C1 ( 1 ) = F n F ex, F C1 ( ) = F ex mlrly or the flux lnked wth conductor t s: F C ( ) = F ex, F C ( 1 ) = F n F ex The self- nd mutul-nductnces re: L 1 = F n 1 M = F ex F ex = L n L ex, L = F ex = L 1 ex = L ex F n F ex B v(t) ~

19 Qus-ttonry Electrodynmcs kn Effect Motvton The two conductors re mgnetclly coupled nductors n the phsor domn descred y: V = R 1 I 1 jw L 1 I 1 jw MI = = R 1 I 1 jw L n I 1 jw L ex I 1 jw L ex I V = R I jw L I jw MI 1 = = R I jw L ex I jw L ex I 1 v(t) ~ - 1 R 1 I 1 jw L n I 1 jw L ex I 1 jw L ex I = = R I jw L ex I jw L ex I 1 1 = E R R 1 jw L n v(t) ~ - R 1 jw M jw L 1 R jw L I = I 1 I Qus-ttonry Electrodynmcs kn Effect Motvton I 1= I R R 1 R 1 jw L n I = I R 1 jw L n R 1 R 1 jw L n v(t) ~ - 1 lm I R 1 = I R ; lm I 1 R 1 R = I R 1 R 1 lm I 1 = 0 ; lm I 1 = I. l l 1 Therefore for lw frequences the current densty s dstruted n the two conductors dependng on the resstnces of ech of them. For hgh frequences the current densty s only n the outer one. v(t) ~ - R 1 jw M jw L 1 R jw L

20 kn Effect Formulton The dstruton of J nsde cylndrcl wre s mxml ner the wre surfce nd decreses t the wre centre so tht the solute vlue of J s. J(r) = J exp(-r/d ) where J s the vlue of J on the surfce. The nductnce of wre s lso ffected y the skn effect. It ehvour s gve n the grph on the sde. The skn depth s depth elow the conductor surfce t whch the the current densty hs fllen to 1/e of J (J 0,37 J ): δ = Qus-ttonry Electrodynmcs E F 1 ωϵ/σ ωϵ/σ J(r)/J L nt /(µ/8p ) d /R = d /R = d /R = r/r d /R Qus-ttonry EM kn Effect Formulton For frequences much elow s /e (n copper f Hz) d s gven y: δ = E F nd for cle length L of dmeter D the resstnce R of t s: f R ŒF( kž)ž The power dsspted y the Joule effect s equl to tht whch would e dsspted f unform current densty were confned to crculr crown of thckness d.

21 Qus-ttonry Electrodynmcs kn Effect Mtgton A type of cle clled ltz wre s used to mtgte the skn effect for frequences of few klohertz to out one meghertz. It conssts of numer of nsulted wre strnds woven together n crefully desgned pttern, so tht the overll mgnetc feld cts eqully on ll the wres nd cuses the totl current to e dstruted eqully mong them. Wth the skn effect hvng lttle effect on ech of the thn strnds, the undle does not suffer the sme ncrese n AC resstnce tht sold conductor of the sme cross-sectonl re would due to the skn effect. Hgh-voltge, hgh-current overhed power lnes often use lumnum cle wth steel renforcng core;; the hgher resstnce of the steel core s of no consequence snce t s locted fr elow the skn depth where essentlly no AC current flows. 41