Circuit Transients time

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1 Circui Tranin A Solp 3/29/0, 9/29/04. Inroducion Tranin: A ranin i a raniion from on a o anohr. If h volag and currn in a circui do no chang wih im, w call ha a "ady a". In fac, a long a h volag and currn ar ady AC inuoidal valu, w can call ha a ady a a wll. Up unil now w'v only dicud circui in a ingl ady a Bu wha happn whn h a of a circui chang, ay from "off" o "on"? Can h a of h circui chang inananouly? No, nohing vr chang inananouly, h circui a will go hrough om raniion from h iniial a, "off" o h final a, "on" and ha chang will ak om amoun of im. Th am i ru in mchanical ym. If you wan o chang h vlociy of a ma or h lvl of fluid in a ank or h mpraur of your coff, ha raniion from on a o anohr will ak om im im Simpl xponnial curv im An xponnial buildup of a in wav Th drawing on hi pag how om ypical ranin ha can occur whn a circui i fir urnd on. Th iniial a of all h wavform i 0. Th final a i ihr or a in wav wih an ampliud of. Noic ha in all four ca h ranin ffc dcay xponnially and ha all four wavform hav pry narly rachd hir ady-a valu by h nd of h graph Ovrhoo im inging im Tranin analyi: Ndl o ay, h analyi of h ranin i a bi mor involvd han h ady a. In fac, i uually involv wo ady a analy ju o find h iniial and final a of h circui, and hn you ill nd o figur ou wha happn in bwn. Tranin ar no inan bcau capacior and inducor in h circui or nrgy, and moving h nrgy ino or ou of h par ak om im. Th volag-currn rlaionhip of capacior and inducor ar diffrnial quaion, o ranin analyi will involv olving diffrnial quaion. Bu don' panic, you'll larn om nic rick and chniqu for daling wih h quaion rick and chniqu ha you can u in any nginring fild, no ju EE. Acually, all ha phaor uff you ud wih AC circui wa alo a rick o implify h diffrnial quaion, unforunaly, ha rick only work for inuoid in ady a. DC circui wih only rior alo xprinc ranin, bu h ar du o non-idal capacianc and inducanc of h par and wir ha w havn' conidrd bfor. Th ranin happn o fa ha w won' worry abou hm. Tranin p..

2 Imporanc: So why ar ranin imporan? Two raon rally. DC and ady-a AC ar fin for moving and uing lcrical powr, bu omim you nd o urn hm on and off and you may nd o know wha happn a ho im. Tha nd urn ou o b rlaivly rar and probably couldn' juify h im w'll pnd udying ranin. I' ignal procing and conrol ym rally driv our udy of ranin. Signal ar lcrical volag and currn ha carry informaion. Th informaion could b audio or vido or h informaion migh b abou h poiion or pd of mchanical par, or abou h mpraur or lvl of fluid or chmical or pracically anyhing you can imagin. To carry informaion ignal hav o chang in om way ha w can' prdic and w'll nd o hav om ida how a circui will rpond o ho chang. Chang ar ranin. Howvr, inc h chang can' b known bforhand w uually find a circui' rpon o pcific yp of inpu and hn draw concluion abou h ffcivn or abiliy in h gnral ca. Ofn h lcrical circui i ju on par of a largr ym ha may includ mchanical, hydraulic, or hrmal ym. S box..2 Fir-ordr ranin Prinr Dign L hink abou om of h ranin and ignal involvd wih moving a prin had and puing ink on a pag of papr. Fir, hr h mchanical ym o mov h prin had. How quickly do h movmn rpond o an lcrical ignal n o h moor? How powrful do ho ignal hav o b? Do i hav a naural frquncy whr i migh vibra of ocilla? Th ar all quion for h ranin analyi of h mchanical ym. Th lcrical circui would ak a ignal from om nor ha indica h poiion of h prin had and, uing ohr informaion abou whr h nx characr hould b prind, nd h righ ignal o h moor. You d u ranin analyi o mak ur ha i could handl any combinaion of inpu wihou ovrhooing h poiion or ocillaing or going oo lowly. Bid hi, h lcrical ym may hav o compna for propri of h mchanical ym. Finally, hr h ym ha acually pu h ink on h papr, l ay i an ink j. Tranin conidraion hr would includ h im i ak for h prin had o ha h ink o h poin whr i pi a bubbl and how ha hould all b imd wih h had movmn o plac ha bubbl on h papr a ju h righ plac. Analyi of a circui wih only on capacior or on inducor rul in a fir-ordr diffrnial quaion and h ranin ar calld fir-ordr ranin. Sri C circui, radiional way: Look a h circui a righ. I how a capacior and a rior conncd o a volag ourc by way of a wich ha i clod a im =0. Bfor h wich i clod h currn i() and h volag v ar boh 0, bu h volag v C i unknown. mmbr a capacior i capabl of oring a charg, o w don' know wha i charg migh b unl w or can maur i or i i givn. I'll call i h iniial volag (v C (0)). Bcau h volag acro a capacior canno chang inananouly, h volag acro h capacior ju afr h wich clo mu b h am a i wa ju bfor h wich clo. Now w ju hav o apply h baic circui law V in = v v. C i v C = i.. C i C d Making h obviou ubiuion. Th nx p hr would b o diffrnial boh id of h quaion, bu if you'r a lil mor clvr, hr' an air way, chck hi ou: Mak hi ubiuion inad i = i C = d d v C, o g V in =. d d v C Waa-laa, no ingraion. Alway ry o wri your diffrnial quaion wihou ingral, i will limina on mor ourc of miak. W now hav a diffrnial quaion in rm of v C. If v C in' h variabl w wan o find in our analyi hn w can alway go back o h circui lar and find h currn or h volag v by impl circui analyi afr w'v found v C. Tranin p..2 v C

3 So now w hav o olv h diffrnial quaion. call from your diffrnial quaion cla h ha fir ordr diffrnial quaion ar alway olvd by quaion of h following form. Sandard fir ordr diffrnial quaion anwr: v C ( ) = A. And, by diffrniaion: Subiu h back ino h original quaion: V in =. d d v C v C =... A. d d v C =.. =.... B. A W can para hi quaion ino wo par, on which i im dpndn and on which i no. Each par mu ill b an quaion. Tim indpndn (forcd) par: V in = A, A = V in = final condiion = Tim dpndn (ranin) par: 0 =... Divid boh id by.. B. o g 0 =., = Thi i calld h "im conan" and will bcom a rahr imporan lil characr. Pu h par w know back ino h xprion for v C ( ) = V. in B a im = 0: v C ( 0 ) = V in B, B = v C ( 0) V in = v C ( 0) And finally: v C ( ) = V. in B = v C ( ) v. C ( 0),. C =, whr =. C. C = v C ( ) B i h diffrnc bwn v C a h ar and v C a h nd. I urn ou ha all fir-ordr ranin oluion will hav h am form, ju diffrn variabl and im conan. Onc you hav v C (), you can alo find v () and/or i() from v C () if you wan. v ( ) = V in v C ( ) = V. in B V in =. B =. B = v. C ( 0) i( ) = d d v C =.. C. C B =. v C ( 0) =. L' plo h and wha hy acually look lik. Th graph how h capacior charging from i' iniial valu o Vin and v falling o 0 (am for i ) V in 63% 37% Th curv ar gnralizd bad on h concp of h im conan, which i why w v C ( 0) im inroducd h im conan. Lar 0 = =2 =3 =4 =5 w'll look a h kind of curv in grar dail. V in v C ( 0) im conan v C () Ok, ha wa fun, bu you migh ak a hi poin if hr in' an air way. Y, in fac, hr i. W'll look a nx. 63% 37% Tranin p..3 0 v () = i() im conan

4 Fir-Ordr Tranin h Eay Way Noic in h prcding analyi ha I mad a vry andard gu a h oluion of h diffrnial quaion. Sandard fir ordr diffrnial quaion anwr: v C ( ) = A. Furhr noic ha A urnd ou o b h final condiion and ha B urnd ou o b h diffrnc bwn h iniial and final condiion. Finally, rmmbr ha I rnamd o -/. All of hi can b gnralizd o any fir ordr ym. Th anwr will alway b in hi form: final condiion / \ --- im conan For all fir ordr ranin: x( ) = x( ) ( x( 0) x( ) ). \ iniial condiion x() could b any variabl in any fir-ordr ym. I could b a mpraur, or a fluid lvl, or a vlociy, bu for u i uually man volag and currn, o w'll hav oluion lik h. v X ( ) = v X ( ) v. X ( 0) v X ( ) or i X ( ) = i X ( ) i. X ( 0) i X ( ) You find Iniial and final condiion from ady-a analyi. Tha lav only on hing ha you hav o find from h diffrnial quaion-- h im conan. If w could only figur ou wha h im conan of a circui (or ym) i, hn w could almo jump raigh o h oluion. Th fir way o find h im conan i o imply rmmbr i' form for a fw ca, lik h for C circui. Evn if h circui don' look xacly lik h andard C ri circui, Thvnin can hlp u mak i look ha way. Sinc narly all of our fir ordr circui will involv a ingl capacior or a ingl inducor hi i no an impracical mhod a all. Anohr way o find h im conan i o manipula h diffrnial quaion ino hi paricular form conan = X. dx d wih no facor in fron of h "X" rm. Whavr h facor in fron of urn ou o b, ha will b. For h C circui h diffrnial quaion could b wrin a V in =. d d v C v C noic ha h facor in fron of d d v C i indd. Finally, hr i an vn air way bad on h LaPlac "" and -impdanc ha w can u in circui and quaion in plac of diffrnial and ingral. You'll hi la mhod lar, afr cond-ordr ranin. (Incidnally, hi i h raon ha I cho o u an a h unknown in h xponnial.) dx d Sri L circui: OK, if i' o ay, l' ry i wih a ri L circui. v in = v v L V in = i. L. d d i V in = i L. d d i So, h im conan mu b = L Tha wan' oo bad. Iniial condiion: i L ( 0 ) = 0 If h wich wa iniially opn h h currn ju bfor h wich wa clod wa 0, and inducor currn can' chang inanly. V in Final condiion: i L ( ) = Th inducor look lik an hor for ady-a D So: i L ( ) = i L ( ) i. V in L ( 0) i L ( ) = Wll, ha' wan' oo painfull, wa i? 0 V. in. L V in =.. L Tranin p..4

5 .3 Iniial and Final Condiion Mor han onc I'v aid ha h iniial and final condiion ar found from ady-a analyi of h circui. I' abou im I aid how. Iniial Condiion: Thr ar wo vry imporan concp ha you u o find h iniial condiion. ) Capacior volag canno chang inananouly, v C (0+) = v C (0-). If you can find h capacior volag ju bfor im = 0 (or whavr ar h ranin), hn you know wha i i ju afr im = 0, v C (0+) = v C (0-). I canno chang inananouly. Ofn you'll u h mhod oulind blow o find h final condiion of h prviou circui, pcially if h circui' bn in ha condiion for "a long im". Somim you'll hav o olv h prviou ranin o find h iniial condiion for h nx ranin. If you canno find h capacior volag ju bfor im = 0 from h circui, hn you'll hav o b old wha h iniial volag or charg i. Capacior can hold a charg for a long im, and can b movd from on circui o anohr wihou loing h charg. High chool lcronic udn lik o charg capacior and lav hm whr hy'll hock om poor unupcing oul. Of cour you'd nvr do omhing a childih a ha. Occaionally you may b old wha h iniial charg i in rm of coulomb. In ha ca rmmbr h dfiniion of capacianc. C = Q V which can b rarrangd o If you hav nohing l o go on, aum h iniial volag i 0. V = Q C 2) Inducor currn canno chang inananouly, i L (0+) = i L (0-). If you can find h inducor currn ju bfor im = 0 (or whavr ar h ranin), hn you know wha i i ju afr im = 0, i L (0+) = i L (0-). I canno chang inananouly. If you canno find h inducor currn ju bfor im = 0 from h circui, hn aum i' 0. al circui and ral inducor alway hav om rianc o inducor currn ju don' la vry long (unl you'r daling wih uprconducor). Inducor would b vry difficul o mov from on circui o anohr wihou loing h currn. If you'r givn an iniial currn for a problm, raliz ha hi i probably ju o mak h problm mor inring, or h iniial currn com from prviou analyi. Do no mix h wo concp up. Capacior currn and inducor volag can boh chang inanly wih no problm a all. Final Condiion: Thi i ady-a analyi. Th ady-a i h final condiion. DC ourc If all h volag and currn ourc ar DC, hn a h final condiion h capacior ar all don don charging o i C = 0, and you can ra hm a opn circui. Whn you find h volag acro h opn, ha will b h final capacior volag. You'v don hi or of hing bfor o find h nrgy ord in a capacior. plac capacior wih opn plac inducor wih wir A h final condiion h inducor currn ar alo no longr changing, o h volag acro an inducor i 0. Tra inducor a wir (hor circui). Whn you find h currn hrough h wir, ha will b h final inducor currn. AC ourc U phaor analyi (jω). mmbr ha phaor analyi wa alo calld "ady-a AC". On of h primary aumpion wa ha h ranin had all did ou. j. ω. C j. ω. L Tranin p..5

6 .4 Exponnial Curv Bfor w go on o cond-ordr ranin w hould ak a clor look a om of h characriic of xponnial curv. Th curv ha how up a anwr o our ranin problm ar hown blow. Th ranin ffc alway di ou afr om im, o h xponn ar alway ngaiv. Ju hink abou wha a poiiv xponn would man. Tha wouldn' b a ranin-- ha would b xponnial growh, lik h populaion. 95% 99% Final condiion 63% Iniial condiion 0% Iniial condiion 00% iing Exponnial Curv im conan, im 37% % Dcaying Exponnial Curv % Final im conan, condiion Som imporan faur: ) Th curv procd from an iniial condiion o a final condiion. If h final condiion i grar han h iniial, hn h curv i aid o b a "riing" xponnial. If h final condiion i l han h iniial, hn h curv i calld a "dcaying" xponnial. 2) Th curv' iniial lop i + /. Ιf hy coninud a hi iniial lop hy'd b don in on im conan. 3) In h fir im conan h curv go 63% from iniial o h final condiion. 4) Afr hr im conan h curv i 95% of h way o h final condiion. 5) By fiv im conan h curv i wihin % of h final condiion and i uually conidrd finihd. Mahmaically, h curv approach h final condiion aympoically and nvr rach i. In raliy, of cour, hi i nonn. Whavr diffrnc hr may b bwn h mahmaical oluion and h final condiion will oon b ovrhadowd by random flucuaion (calld noi) in h ral circui. Tranin p..6