Circuitos RL, LC y RLC

J@S@CN @@ CNT@\_BG Q@@@ Q@@@+ C@UC_@\GS UB+BC P BUC Q@@@(> + C`rcu`tg ros`st`vg – `hluct`vg !UB) So nhnbz`n n cght`hunc`óh cght`hunc`óh uh c`rcu`tg quo pgsoo uhn ros`stohc`n obàctr`cn U oh sor`o cgh uhn `hluctnhc`n B" s`ohlg lo `htoràs ostul`nr bgs trnhs`tgr`gs quo so prglucoh nb corrnr y nb ner`r ob `htorruptgr lo osto c`rcu`tg(

J`furn >(>3 Nb corrnr ob `htorruptgr so ostneboco uhn cgrr`ohto obàctr`cn quo vn croc`ohlg n trnvàs lob t`okpg y quo pgr osn cnusn fohorn fohorn uhn jok `hluc`ln oh bn `hluctnhc`n" bn quo so gpgho nb croc`k`ohtg lo l`can cgrr`ohto" hg gestnhto bg cunb bn cgrr`ohto cght`hùn `hcrokohtîhlgso anstn su vnbgr kîx`kg" oh ob cunb so knht`oho cghstnhto( So pbnhton oh pr`kor bufnr bn ocunc`óh lo knbbns lob c`rcu`tg lo bn J`furn >(>3 cgh ob prgpós`tg lo losnrrgbbnr bns oxpros`ghos kntokît`cns quo r`foh bn ovgbuc`óh lo bn cgrr`ohto obàctr`cn oh ob t`okpg lurnhto bgs ostnlgs trnhs`tgr`gs lob k`skg( Zr ; ο -

ο`

`(U ; ο + B(l`'lt `(U - B(l`'lt ;

ο

>(>3

Uosubtn uhn ocunc`óh l`jorohc`nb quo kol`nhto uh cnke`g lo vnr`nebos so trnhsjgrkn oh uhn ocunc`óh lo vnr`nebos sopnrnebos cuyn rosgbuc`óh os sohc`bbn( L`v`l`ohlg k`okerg n k`okerg pgr U bn oxpros`óh >(>3 y rogrlohnhlg9 !B'U)(!l`'lt) !B'U)(!l`'lt) ; !ο 'U) + ` S` so loj`ho9 x ; !ο 'U) – ` Uosubtn9 lx ; + l`

5

Buofg bn ocunc`óh lo knbbn >(>3 so puolo oxprosnr oh juhc`óh lo x cgkg s`fuo9 +!B'U)(!lx'lt) ; x Cuyn rosgbuc`óh os bn s`fu`ohto9 lx'x ; + lt' !B'U) lx'x ; +

lt' !B'U)

bh !x'xg) ; + t'!B'U) x'xg ; o

+ t'!B'U)

Uookpbnznhlg x pgr su loj`h`c`óh oh juhc`óh lo bn vnr`nebo gr`f`hnb" bn cgrr`ohto `9 x ; !ο 'U) – ` P hgtnhlg quo9 xg ; !ο 'U) yn quo n t ; 4 bn cgrr`ohto ` os os hubn Uosubtn9 + t'!B'U) !ο 'U) – ` ; x ; !ο 'U)( o ` ; !ο 'U) – !ο 'U) o ` ; !ο 'U)(!> + o

+ t'!B'U)

+ t'!B'U)

)

5(>3

Nb cgc`ohto B'U" quo t`oho uh`lnlos lo t`okpg" sofuhlgs oh ob S@" so bg lohgk`hn cghstnhto lo t`okpg lob c`rcu`tg UB " bn cunb so s`kegb`zn όb( Ob s`fu`ohto frîj`cg lo bn oxpros`óh nhtor`gr kuostrn quo9 `;4 pnrn t ; 4 ` ; ο 'U pnrn t  `hj`h`tg Os loc`r" quo pnrn t`okpg `funb g knygr n c`hcg cghstnhtos lo t`okpg" ? όb" bn cgrr`ohto os `funb n bn quo tohlrën ob c`rcu`tg s`h bn `hluctnhc`n" os loc`r" so cgkpgrtn cgkg ros`st`vg purg" bg cunb so loeo n quo bn cgrr`ohto yn hg vnrën oh ob t`okpg y pgr bg tnhtg hg any prosohc`n lo jok `hluc`ln(

3

J`furn 5(>3 So cgkpruoen nsë bg nht`c`pnlg cunb`tnt`vnkohto oh ob soht`lg lo quo bn `hluctnhc`n rotnrln ob croc`k`ohtg lo bn cgrr`ohto" porg quo j`hnbkohto àstn so ostneboco oh ob vnbgr quo cgrrospghlo bn jok" ќ" y bn ros`stohc`n U( N knygr cghstnhto lo t`okpg knygr os ob rotnrlg pnrn nrr`enr n bn cgrr`ohto j`hnb(

J`furn 3(>3 S` uhn voz ostneboc`ln bn cgrr`ohto j`hnb so ronb`zn bn nporturn lob `htorruptgr y so cghoctn rîp`lnkohto cgkg `hl`cn bn J`furn <(>3" bn juohto quoln n`sbnln k`ohtrns quo ob c`rcu`tg porknhoco corrnlg oh uhn knbbn quo cght`oho sóbg n U y n B(

J`furn 5?(0

<

So vor`j`cn quo bn cgrr`ohto cgk`ohzn n l`sk`hu`r bg quo fohorn uhn jok `hluc`ln quo so gpgho nb cnke`g quo oh osto cnsg os bn l`sk`huc`óh" pgr bg quo rotnrln ob nrr`eg n corg lo bn cgrr`ohto( Nsë cgkg pnrn ob c`orro lob `htorruptgr bn `hluctnhc`n so gpgho n quo bn cgrr`ohto nukohto" oh osto cnsg so gpgho n quo l`sk`huyn( Loslo gtrg puhtg lo v`stn so vor`j`cn quo bn ohorfën quo so hocos`tó pnrn cronr ob cnkpg knfhàt`cg nb corrnr ob `htorruptgr" k`ohtrns bn cgrr`ohto crocën b`k`tnln pgr ob ojoctg lo bn `hluctnhc`n " nb cghoctnr ob c`rcu`tg sofùh bn J`furn <(>3" osn ohorfën os lovuobtn y ob cnkpg cronlg vn losnpnroc`ohlg" pgr bg cunb bn cgrr`ohto hg l`sk`huyo `hstnhtîhonkohto n corg( Cneo ncgtnr quo s` ob c`rcu`tg juorn ros`st`vg ros`st`v g purg bn cgrr`ohto obàctr`cn nukohtnrën `hstnhtîhonkohto n su vnbgr j`hnb nb corrnr ob `htorruptgr" y cnorën `hstnhtîhonkohto n corg oh su nporturn(

J`furn ?(>3 Tnrn ob c`rcu`tg lo bn J`furn <(>3 bn ocunc`óh lo knbbn rosubtn9 Zr ; ο `(U ; + B(l`'lt Ocunc`óh l`jorohc`nb lo vnr`nebos sopnrnebos" cuyn rosgbuc`óh os sohc`bbn( l`'` ; + lt' !B'U) l`'` ; + lt' !B'U)

? bh !`'@g) ; + t'!B'U) ` ; @g(o

+ t'!B'U)

Cneo lostncnr quo bn cgrr`ohto `h`c`nb pnrn ostn cghox`óh rosubtn `funb n bn kîx`kn j`hnb lo bn cghox`óh sofùh bn J`furn >(>3" @kîx ; ο 'U( ` ; !ο 'U)(o

+ t'!B'U)

3(>3

Lghlo9 όb ; B'U Bn oxpros`óh 3(>3 loscr`eo bn pnrto lo bn J`furn :(>3 oh quo bn cgrr`ohto ` l`sk`huyo( So cghs`lorn quo bn cgrr`ohto sorî prîct`cnkohto hubn n t ; ?( όb(

J`furn :(>3 Q@@@(5 + C`rcu`tg `hluct`vg – cnpnc`t`vg !BC) Gsc`bnc`ghos obàctr`cns _h c`rcu`tg cghst`tu`lg sgbnkohto pgr uhn `hluctnhc`n y uh cnpnc`tgr" y cuyn ros`stohc`n so losproc`n" so ohcuohtrn `h`c`nbkohto cgh ob `htorruptgr ne`ortg y ob cnpnc`tgr cnrfnlg pgr uhn juohto quo so an rot`rnlg" s`ohlg bn cnrfn lob cnpnc`tgr9 ^ ; C( ο Lghlo9 ^9 cnrfn ncukubnln C9 os ob vnbgr lo bn cnpnc`lnl ο9 vnbgr lo bn jok cgh quo so cnrfó ob cnpnc`tgr

:

J`furn 6(>3 Nb corrnr ob `htorruptgr cgk`ohzn n c`rcubnr uhn cgrr`ohto obàctr`cn snb`ohto lob pgs`t`vg lob cnpnc`tgr" os loc`r" ob cnpnc`tgr so loscnrfn( Ostn cgrr`ohto obàctr`cn vn croc`ohlg n trnvàs lob t`okpg pgr bg cunb su ostneboc`k`ohtg ostî s`ohlg rotnrlnlg pgr bn `hluctnhc`n(

J`furn 1(>3 S` bn `hluctnhc`n hg ostuv`orn prosohto ob cnpnc`tgr so loscnrfnrën kuy rîp`lnkohto" sus cnrfns pgs`t`vns y hofnt`vns so nhubnrënh kutunkohto" nrr`enhlg n corg( Bn cgrr`ohto croco anstn anstn uh vnbgr kîx`kg oxnctnkohto cunhlg bn cnrfn oh ob cnpnc`tgr os corg(

J`furn 0(>3 N pnrt`r lo quo bn cnrfn oh ob cnpnc`tgr os hubn bn cgrr`ohto cgk`ohzn n l`sk`hu`r lo kglg quo bn jok `hluc`ln oh bn `hluctnhc`n oh osto cnsg so gpgho n osn l`sk`huc`óh" tnb quo cght`hùn ane`ohlg cgrr`ohto obàctr`cn oh ob k`skg soht`lg y ob cnpnc`tgr so cnrfn oh soht`lg cghtrnr`g anstn ob k`skg vnbgr lo cnrfn quo nb `h`c`g y nrr`enhlg bn cgrr`ohto n corg((

6

J`furn >4(>3 P cgk`ohzn huovnkohto uhn cgrr`ohto croc`ohto nagrn agrnr`n lo kglg s`k`bnr nb nhtor`gr" anstn quo ob cnpnc`tgr so cnrfn huovnkohto cgkg nb `h`c`g( So trntn lo uhn gsc`bnc`óh obàctr`cn " s`h pàrl`ln lo ohorfën pgrquo so an cghs`lornlg uhn s`tunc`óh `lonb lo ros`stohc`n hubn" lghlo bn cgrr`ohto obàctr`cn rosubtn sucos`vnkohto nht`agrnr`n y agrnr`n pnsnhlg lo vnbgr corg n kîx`kg" n corg" n kîx`kg oh soht`lg cghtrnr`g" n corg" y nsë sucos`vnkohto( Cneo ncgtnr quo cunhlg bn cgrr`ohto vn croc`ohlg so ostneboco uh cnkpg knfhàt`cg quo os kîx`kg cunhlg bn cgrr`ohto os kîx`kn y bn cnrfn oh ob cnpnc`tgr os hubn( Osto cnkpg knfhàt`cg so jgrkn lurnhto ob croc`k`ohtg lo cgrr`ohto y n pnrt`r lo @kîx y q ; 4 os bn ohorfën ncukubnln cgkg cnkpg knfhàt`cg bn quo sustohtn bn cght`hu`lnl lo bn cgrr`ohto oh ob k`skg soht`lg( Bn ohorfën so vn trnhsj`r`ohlg sucos`vnkohto lob cnpnc`tgr n bn `hluctnhc`n( Oh bn J`furn >>(>3 so rocgp`bn bg loscr`ptg anstn nquë pnrn ob c`rcu`tg BC(

J`furn >>(>3 Bn ocunc`óh lo knbbn os9 Zc ; ο` q ' C ; + B(l`'lt B(!l`'lt) - q ' C ; 4 P cgkg9

1 5

` ; lq'lt rosubtn l`'lt ; l q'lt

5

Buofg9 5

5

B( !l q'lt )- q'C ; 4

<(>3

Bns gsc`bnc`ghos obàctr`cns rosubtnh oqu`vnbohtos n bns gsc`bnc`ghos kocîh`cns lo uhn knsn suiotn n uh kol`g obîst`cg(

J`furn >5(>3 Bn juorzn lo rost`tuc`óh os9 J ; +M(x Lghlo9 J9 juorzn M9 cghstnhto obîst`cn x9 l`stnhc`n quo ob rosgrto so npnrtn lob oqu`b`er`g tnhtg oh cgkpros`óh cgkg oh ost`rnk`ohtg( Cgkg os bn ùh`cn juorzn prosohto" rosubtn9 Θ

J ; J ; +M(x ; k(n

P toh`ohlg oh cuohtn quo9 5

5

n ; lv'lt ; l x'lt Buofg9

0

5

5

+M(x ; k(!l x'lt ) Uogrlohnhlg9 5

5

k(!l x'lt ) - M(x ; 4

?(>3

Ocunc`óh oqu`vnbohto n bn cgrrospghl`ohto n bn gsc`bnc`óh l c`rcu`tg BC" <(>3( Bns oqu`vnbohc`ns ohtro bns vnr`nebos lo bns ocunc`ghos cgrrospghl`ohtos nb c`rcu`tg lo gsc`bnc`ghos obàctr`cns" <(>3" y nb s`stokn lo gsc`bnc`ghos kocîh`cns" ?(>3" sgh bns quo so kuostrnh n cght`hunc`óh( Gsc`bnc`ghos obàctr`cns

Gsc`bnc`ghos kocîh`cns

B q >'C

k x M

Sne`ohlg quo bn sgbuc`óh lo bn ocunc`óh ?(>3 os9 x ; xg ( cgs π(t

:(>3

P π ; M'k

6(>3

Tgr nhnbgfën so cghcbuyo quo pnrn ob c`rcu`tg lo gsc`bnc`ghos obàctr`cns9 q ; ^kîx ( cgs π(t

1(>3

^kîx ; C( ο π

; >' B(C

0(>3

P lnlg quo9 ` ; lq'lt Uosubtn9 ` ; + ^kîx( π ( soh π(t @kîx ; ^kîx(

>4(>3

π

Buofg9 ` ; + @kîx( soh π(t

>>(>3

>4 Frnj`cnhlg Frnj`cnhlg q!t) o `!t)9

J`furn >3(>3 5

Cunhlg bn cnrfn os ^kîx" ` ; 4 y _ ; !^kîx)  '5(C" tgln bn ohorfën ohorfën so ohcuohtrn ncukubnln oh ob cnpnc`tgr( 5 Cunhlg bn cgrr`ohtos os @kîx" q ; 4 y _ ; B(!@kîx)  '5" tgln bn ohorfën ohorfën so ohcuohtrn ncukubnln oh bn `hluctnhc`n( Bn ohorfën so knht`oho cghstnhto lurnhto bn gsc`bnc`óh obàctr`cn cgkpbotn" pgr bg quo os hocosnr`g lokgstrnr quo bn ohorfën kîx`kn ncukubnln oh ob cnpnc`tgr y bn ohorfën kîx`kn ncukubnln oh bn `hluctnhc`n sgh `funbos( Cgkg @kîx ; ^kîx(

π

5

5

5

_ ; B(!^kîx) ( π '5 ; B(!^kîx) ( !>'BC) '5 ; !^kîx) ' 5(C

>5(>3

Oh fohornb bn ohorfën ncukubnln oh juhc`óh lob t`okpg sorî9 _t ; _b - _c 5

5

_t ; !B(` ' 5) - !q ' 5(C) 5

5

5

_t ; !B'5)(!@kîx) !B'5)(!@kîx) ( π (soh

π(t

5

5

- !>'5(C)(!^kîx) !>'5(C)(!^kîx) (cgs

π(t

>3(>3

>>

J`furn ><(>3 Q@@@(3 + C`rcu`tg `hluct`vg" cnpnc`t`vg" ros`st`vg !BUC) S` `hcgrpgrnkgs nb c`rcu`tg BC uhn ros`stohc`n U bns gsc`bnc`ghos obàctr`cns so vorîh njoctnlns pgr bn pàrl`ln lo ohorfën quo `kpb`cn bn ros`stohc`n" pgr bg cunb bns gsc`bnc`ghos sorîh nkgrt`funlns g `hcbus`vo puolo hg aneor gsc`bnc`óh( Huovnkohto ox`sto sokoinhzn cgh bns gsc`bnc`ghos kocîh`cns nkgrt`funlns(

J`furn >?(>3 Bn ocunc`óh lo knbbns rosubtn nagrn9 Zc - Zr ; ο` q ' C - `(U ; + B(l`'lt Uogrlohnhlg y toh`ohlg oh cuohtn quo9 5

5

` ; lq'lt" lq'lt" rosubtn l`'lt ; l q'lt Buofg9 5

5

B( !l q'lt )- U(!lq'lt) - q'C ; 4

><(>3

Ostns gsc`bnc`ghos obàctr`cns nkgrt`funlns rosubtnh oqu`vnbohtos n bns gsc`bnc`ghos kocîh`cns nkgrt`funlns(

>5

J`furn >:(>3 Bn gsc`bnc`óh kocîh`cn nkgrt`funln os bn quo gcurro" pgr bg quo so loeo cghs`lornr quo nlokîs lo bn juorzn lo rost`tuc`óh obîst`cn cohtrnb`znln ox`sto uhn juorzn lo rgznk`ohtg quo os prgpgrc`ghnb n bn vobgc`lnl lo bn knsn gsc`bnhto" v ; lx'lt( Θ

J ; J - Jr ; k(n

P toh`ohlg oh cuohtn quo9 5

5

n ; lv'lt ; l x'lt P quo9 Jr ; + e(!lx'lt) Uosubtn9

5

5

+M(x + e(!lx'lt) ; k(!l x'lt ) Uogrlohnhlg9 5

5

k(!l x'lt ) - e(!lx'lt) - M(x ; 4

>?(>3

Ocunc`óh oqu`vnbohto n bn cgrrospghl`ohto n bn gsc`bnc`óh l c`rcu`tg BUC( Oqu`vnbohc`n ohtro bns vnr`nebos lo bns ocunc`ghos cgrrospghl`oht cgrrospghl`ohtos os nb c`rcu`tg lo gsc`bnc`ghos obàctr`cns y nb s`stokn lo gsc`bnc`ghos kocîh`cns( Gsc`bnc`ghos obàctr`cns B q >'C U

Gsc`bnc`ghos kocîh`cns k x M e

>3

Sne`ohlg quo bn sgbuc`óh lo bn ocunc`óh >?(>3 os9 x ; xg ( o P9

πl

;

–t'5(!k'e)

( cgs πl(t

>:(>3 5

!m'k) – !e'5(k)

>6(>3

Tgr nhnbgfën so cghcbuyo quo pnrn ob c`rcu`tg lo gsc`bnc`ghos obàctr`cns9 q ; ^kîx (o

–t'5(!B'U)

( cgs πl(t

>1(>3

–t'5(!B'U)

–5t'!B'U)

^kîx (o os bn nkpb`tul" quo nagrn `hcbuyo ob jnctgr quo `hcbuyo o " ob cunb vn l`sk`huyohlg cgh ob t`okpg" bg cunb so vor`j`cn oh bn J`furn >6(>3" lghlo so puolo gesorvnr pgr oiokpbg cgkg l`sk`huyoh bgs vnbgros kîx`kgs lo q n trnvàs lob t`okpg( S`ohlg9 πl

;

!>'BC) – !U'5(B)

5

>0(>3

J`furn >6(>3 Tnrn ob c`rcu`tg BUC ox`sto uh vnbgr lo ros`stohc`n" lohgk`hnlg ros`stohc`n crët`cn " pnrn ob cunb ob rnl`cnhlg lo bn ocunc`óh >0(>3 rosubtn hubg y pgr bg tnhtg πl ; 4( Tnrn vnbgros kohgros lo U rospoctg lo bn ros`stohc`n crët`cn πl 8 4 y ob s`stokn os gsc`bnhto nkgrt`funlg sofùh bn curvn > lo bn J`furn >6(>3( S` πl ; 4" pnrn ob vnbgr lo ros`stohc`n crët`cn" ob kgv`k`ohtg so lohgk`hn nkgrt`funlg y hg gcurro h`hfuhn gsc`bnc`óh" curvn 5( S` πl os `knf`hnr`n" puos bn rnëz os hofnt`vn" ob kgv`k`ohtg os sgeronkgrt`funlg y tnkpgcg gcurro gsc`bnc`óh nbfuhn" curvn 3( Bg nhtor`gr os vîb`lg tnhtg pnrn bns gsc`bnc`ghos obàctr`cns cgkg pnrn bns kocîh`cns(