Lecture 1: Growth and decay of current in RL circuit. Growth of current in LR Circuit. D.K.Pandey

Transcription

1 cur : Growh and dcay of currn in circui Growh of currn in Circui us considr an inducor of slf inducanc is conncd o a DC sourc of.m.f. E hrough a rsisr of rsisanc and a ky K in sris. Whn h ky K is swichd on, h currn in circui sard o incras. Th currn in h circui dos no aain h maximum sady sa valu (E/) a onc bcaus h inducd.m.f. producd across h inducor opposs h growh d of currn. Hnc h currn in h circui incrass slowly o aain is sady sa valu. According o KV, h algbraic sum of insananous volag drop across h circui lmns for a closd loop is zro. Thus, for h prsn circui, E V E V + V V () f a any insan h currn in circui is hn h ponial drop across rsisr and inducor will b and rspcivly. d Using q.(), w can wri, E + d E d d () E ngraing q.() d + C E log ( E ) + C (3) Hr C is ingraion consan and is drmind by iniial condiion. i.. whn, hn from q.(3) w hav log C (4) From qs.(3) and (4), w can wri, log ( E ) log Fig log ( E ) + log log ( E ) log ( E ) log (E) E E E E E () D.K.Pandy Fig. Do no publish i. Copy righd marial.

2 cur : Growh and dcay of currn in circui Sinc, h sady sa currn or maximum currn ( ) E/ And dimnsion of / dimnsion of im; Say, induciv im consan Hnc q.(4) bcoms as, (6) Eqs. () and (6) ar calld as xprssion of growh currn in circui. Ths xprssions indicas ha- () niially h currn in h circui is zro. () incrass xponnially following xprssion ( ). (3) Afr infini im currn rachs o is sady sa valu ( ). Do no publish i. Copy righd marial. nduciv Tim consan Th Growh of currn in circui follows following quaion. { / } Whn hn from abov q., w can wri { }.63 { (/ ) } { (/.78) } {.368} % Dcay of currn in Circui Thus h im consan for h circui is h im in which h currn incrass up o 63.% of maximum currn. us considr a chargd inducor of slf inducanc is conncd o a rsisr of rsisanc hrough a ky K in sris. Whn h ky K is swichd on, h inducor dischargs hrough rsisr. Th currn in circui sard o dcras du o loss of induciv nrgy hrough rsisr. According o KV, h algbraic sum of insananous volag drop across h circui lmns for a closd loop is zro. Thus E V V E V + V Hr E, Thus V +V () Fig. f a any insan h currn in circui is ngraing q.() hn h ponial drop across rsisr and d C + inducor will b and rspcivly. d log + C (3) Using q.() Hr C is ingraion consan and is + d drmind by iniial condiion. i.. whn, hn from q.(3) w hav d log C (4) d () From qs. (3) and (4), w can wri, D.K.Pandy

3 l og + log log cur : Growh and dcay of currn in circui Do no publish i. Copy righd marial. 3 () Sinc, dimnsion of / dimnsion of im; Say, im consan Hnc q.() bcoms as, / (6) Eqs. () and (6) ar calld as xprssion for dcay of currn in circui. Ths xprssions indica ha niially h currn has maximum valu and dcrass xponnially following xprssion ( ). Fig. nduciv Tim consan Th xprssion for dcay of currn in circui is / Whn hn from abov q., w can wri % Thus h im consan for h circui is h im in which h currn dcays from sady sa valu o 36.8% of maximum currn. No A: n h following circui, whn A is conncd B hn currn riss in h circui and inducor chargs hrough rsisr. Furhrmor, whn A is conncd o C hn h chargd inducor dischargs hrough rsisr. Ω H E V A S Ky K n h Givn circui, Maximum currn E / /. amp Tim consan / / sc Th calculad growh and dcay of currn in h givn circui ar prsnd in h Tabl A and is shown in Fig.A. Tabl A Fig. A Currn (amp) (sc) growh (amp) dcay (amp) Tim (sc) C B Growh Dcay D.K.Pandy Th abov Tabl and Figur indica ha whn hn h growh currn incrass up o approximaly maximum valu and dcay currn dcrass up o zro valu.

4 cur : Growh and dcay of currn in circui No B: hnry vol Uni of ohm (amp/sc) ohm vol. sc vol. sc sc amp.ohm vol Dimnsion of for circui dimnsion of im No C: a of currn growh in circui / Sinc ( ) d Hnc h growh of currn in circui is Th ngaiv sign indicas ha currn dcays wih im.. i... f : small hn currn growh: fas. f : larg hn currn growh: slow No D: a of currn dcay in circui Sinc d Hnc h growh of currn in circui is invrsly proporional o im consan. Th ngaiv sign indicas ha currn dcays wih im... f : small hn currn growh: fas. f : larg hn currn growh: slow No E: Boh growh and dcay of currn in circui is invrsly proporional o im consan. Exampl : Th im consan of an inducanc coil is.x -3 sc. Whn 6 Ω rsisanc is addd in sris, h im consan rducs o.x -3 sc. Find h inducanc and rsisanc of coil. Soluion: Th im consan for circui is and afr addiion of 6 Ω rsisanc, i bcoms. () () + 6 From qs.() and (), w hav Ω Using q.() hnry Exampl : An inducor of inducanc hnry and a rsisr of rsisanc 3Ω is conncd o a d.c. sourc in sris. Find h im in which h currn rachs o half of maximum currn in h circui. Soluion: Givn ha, hnry, 3 Ω, / and? / ( ) / ( ) log D.K.Pandy Do no publish i. Copy righd marial. 4

5 cur : Growh and dcay of currn in circui log.36 log sc Exampl 3: An inducor of inducanc 4hnry and a rsisr of rsisanc Ω is conncd o a d.c. sourc of 6vols. Find h currn afr 4 sc. Soluion: Givn ha, E6 vols, 4 hnry, Ω, 4sc,? E / 6 4 ( ) ( 4/ ) ( ) (.368) amp Exampl 4 n an circui wih sourc, h currn rachs o on hird of is maximum valu wihin sc. Find h im consan of h circui. Soluion: Givn ha, /3,? / ( ) / ( ) 3 / ( ) 3 / / log.36 log.36 ( log 3 log ).36 (.477.3).36 (.76) sc.4.33 sc Do no publish i. Copy righd marial. Exampl A rsisr of Ω and an inducor of 4hnry is conncd o sourc of vol in sris. Find h im in which currn in circui bcoms amp. Soluion: Givn ha, Ω, 4hnry, Evols amp,? E / ( ) / 4 ( ) / 4 / 4 / 4 log.36log sc Exampl 6 A chargd inducor of 4hnry dischargs hrough a rsisr of Ω. Find h im in which currn dcays o 36.8% of is maximum currn. Soluion: Givn ha, Ω, 4hnry, f 36.8%,? W know ha whn 36.8%, / 4 / 8 8 sc Exampl 7 Th im consan for a circui is sc. n cas of dcay of currn, find h im in which currn dcays o half of is maximum. Soluion: Givn ha, sc, /,? / / / / log.36log sc D.K.Pandy