cur 4&: R circui harging of capacior hrough inducor and rsisor us considr a capacior of capacianc is conncd o a D sourc of.m.f. E hrough a rsisr of rsisanc R, an inducor of inducanc and a y K in sris. Whn h y K is swichd on, h charging procss of capacior sars insananous currn. Th charg on capacior incrass wih im and aains is maximum in crain duraion of im. s h im passs, h charg on capacior incrass and gs is maximum. Th naur of charg incras dpnds on h valu of inducor and rsisor. Th naur of charg incras can b analyzd in h following ways. ccording o KV, h algbraic sum of insananous volag drop across h circui lmns for a closd loop is zro. Thus E V V V3 E V V () V3 Fig. If h currn in circui a im is i and d d charg sord on capacior is hn h ponial drop ω ( ) across rsisor, capacior and inducor will b Ri, /, x di hn and rspcivly. d x dx ω x (4) Using.() di E Ri Th.(4) is diffrnial uaion for growh charg across h plas of capacior in R R d d E circui. onsidr h soluion of.(4) is: x () Diffrniaing.() w.r.. im w hav: d R d E () dx x Sinc dimnsion of / is T - and dimnsion of /R is T. Thus w can considr ha: d x R ω and x τ dx d x Puing valu of and in.(4) w g, Hr ωis angular fruncy of oscillaion. nd τ is im consan. Th consan is calld as damping consan. x x ω x Now.() bcoms as; ω (6) d d E ω E. (6) provids ha, 4 4ω d d E ω ω (3) ω Thus consan has wo valus, say hy E E Sinc, E, hus.(3) bcoms as, ar and hn. ω Do no publish i. opy righd marial. Dr. D. K. Pandy
cur 4&: R circui ω and ω Sinc has wo valus hus soluion of.(4) can b wrin as linar combinaion of and. x (7) Diffrniaing.(7) w.r.o im w g: d I (8) Th consans and can b drmind by applying iniial condiion. Th iniial condiions ar: d, and I Undr hs condiions h s.(7) and (8) provids ha, (9) nd ω ω ( ) ω ( ) ω ( ) ( ) ω () Solving s. (9) and (), w hav: ω ω Puing valus,, and in.(7), w hav: ω ( ω ) ( ω ) ω () Th.() is h gnral xprssion for growh of charg across h plas of capacior in R circui. This uaion can b analyzd in h rms of following hr cass. as: Havily dampd or dad ba or ovr dampd condiion: Th condiion, in which h damping rm is grar han h oscillaory rm, is calld as ovr dampd condiion. R i.. > ω or > or R > In his condiion, ω posiiv < Hnc, ω ngaiv Thrfor h scond and hird RHS rms of.() dcay xponnially and bcoms ual o zro a. s a rsul approachs o. i.. In h dad ba condiion, h charg on capacior incrass wih im and afr i gains is maximum valu (Fig.). as: riically dampd condiion: Th condiion, in which h damping rm is approximaly ual o h oscillaory rm, is calld as criically dampd condiion. R i.. ω or or R In his condiion, ω posiiv h Hnc, ω h So, Th.(7) bcoms as, ( h) h ( h) h { } Do no publish i. opy righd marial. Dr. D. K. Pandy
cur 4&: R circui { ( h) (- h) } {( ) ( ) h} ω h h { } h () Th scond rm of.() show h fas dcay of charg and bcoms ual o zro in a small im. s a rsul, h charg on apacior incrass rapidly wih im and i gains is maximum valu in vry soon im (Fig.). as3: ighly dampd condiion: Th condiion, in which h damping rm is lss han h oscillaory rm, is calld as lighly dampd or dampd harmonic condiion. i.. Thus, < ω or R < or R < ω ( ω ) jβ Hnc, ω So, Th.() bcoms as, j ( β) jβ ( jβ) jβ jβ jβ jβ jβ jβ j j β β jβ jβ β j cos β sin β (3) β sin φ an φ β cos φ β β Thn cos φ β ω Hnc.(3) bcoms as, sin φ cos β sin β cos φ cos β cos φ sin β sin φ cos φ ω cos ( β - φ) β { } ω cos ( β - φ) (4) β Th scond rm of.(4) show xponnial harmonic dcay. Thus h charg on capacior incrass wih im and i oscillas abou whos ampliud dcrass xponnially. fr i sauras o is maximum valu (Fig.). Fig. Do no publish i. opy righd marial. 3 Dr. D. K. Pandy
cur 4: R circui Discharging of capacior hrough inducor and rsisor us considr a chargd capacior of capacianc is conncd o a rsisr of rsisanc R, an inducor of inducanc and a y K in sris. Whn h y K is swichd on, h discharging procss of capacior sars hrough inducor and rsisor. Th charg on capacior dcrass wih im du o loss of nrgy hrough rsisor. Th naur of charg dcras dpnds on h valu of inducor and rsisor which can b analyzd in h following ways. ccording o KV, h algbraic sum of insananous volag drop across h circui lmns for a closd loop is zro. Thus E V V V 3 V V V3 V V3 E s E, V () If h currn in circui a im is i and charg rmaind on capacior is hn h ponial drop across rsisor, capacior and inducor will b Ri, / and Using.() di Ri di rspcivly. R d d d R d () Sinc dimnsion of / is T - and dimnsion of /R is T. Thus w can considr ha: R ω and τ hr ωis angular fruncy of oscillaion. nd τ is im consan. Th consan is calld as damping consan. Now.() bcoms as; d d ω (3) Th.(3) is diffrnial uaion for dcay of charg across h plas of capacior in R circui. onsidr h soluion of.(3) is: (4) Diffrniaing.(4) w.r.. im w hav: d Fig. d d d Puing valu of and in.(3) w g, ω ω () E. () provids ha, 4 4ω ω Thus consan has wo valus, say hy ar and hn. ω ω and Sinc has wo valus hus soluion of.(3) an b wrin as linar combinaion of and. (6) Diffrniaing.(6) w.r.o im w g: d I (7) Th consans and can b drmind by applying iniial condiion. Th iniial condiions ar: Do no publish i. opy righd marial. 4 Dr. D. K. Pandy
cur 4: R circui d, and I Undr hs condiions h s.(6) and (7) provids ha, (8) nd ω ω ( ) ω ( ) ω ( ) ( ) (9) ω Solving s. (8) and (9), w hav: ω ω Puing valus,, and in.(6), w hav: ( ) ω ω ( ω ) ω () Th.() is h gnral xprssion for dcay of charg across h plas of capacior in R circui. This uaion can b analyzd in h rms of following hr cass. as: Havily dampd or dad ba or ovr dampd condiion: Th condiion, in which h damping rm is grar han h oscillaory rm, is calld as ovr dampd condiion. Do no publish i. opy righd marial. R i.. > ω or > or R > In his condiion, ω posiiv < Hnc, ω ngaiv Thrfor boh h RHS rms of.() dcay xponnially and nds o zro a. Thus, in h dad ba condiion, h charg on capacior dcrass xponnially wih im and afr i nds o zro. (Fig.). as: riically dampd condiion: Th condiion, in which h damping rm is approximaly ual o h oscillaory rm, is calld as criically dampd condiion. R i.. ω or or R In his condiion, ω posiiv h Hnc, ω h So, Th.(6) bcoms as, ( h) ( h) h h { } ( h) ( - ( ) ( ) h ω { h) } { h} h h { } () Th.() show ha h charg on capacior dcrass vry fas wih im du o rm and i nds o zro in vry soon im (Fig.). as3: ighly dampd condiion: Th condiion, in which h damping rm is lss han h oscillaory rm, is calld as lighly dampd or dampd harmonic condiion. Dr. D. K. Pandy
cur 4: R circui i.. Thus, < ω or R < or R < ω ( ω ) jβ Hnc, ω So, Th.() bcoms as, j ( β) jβ ( jβ) jβ jβ jβ Do no publish i. opy righd marial. jβ j j β β jβ jβ jβ jβ β j cos β sin β () β sin φ an φ β cos φ β β Thn cos φ β ω Hnc.() bcoms as, sin φ cos β sin β cos φ { cos β cos φ sin β sin φ} cos φ ω cos ( β - φ) (3) β Th.(3) shows ha h Thr is xponnial harmonic dcay of charg on capacior wih im. i.. harg on capacior oscillas abou whos ampliud dcays xponnially. fr i nds o zro (Fig.). Thus, hr occurs a 6 dampd harmonic oscillaion of charg on capacior in cas of ligh damping. Fig. No: Th ualiy facor for R circui undr lighly dampd condiion is ωτ ω/r. NoB: If f is h fruncy of oscillaion of charg undr lighly dampd condiion hn, Sinc β ω R πf f π R 4 Qu: capacior of μf, an inducor of.8mh and a rsisor of Ω ar conncd in sris. Is h circui is oscillaory? If ys, hn calcula is fruncy. ns: Givn ha, μf,.8mh, R Ω 3.8 6 4 Dr. D. K. Pandy 4 Ω Sinc R < hus h circui is oscillaory. f π () 3 6 3.8 4 (.8 ) 3.73 f π 6 64 π 64 3.4 8.73 3 f.34 Hz 3.4 Hz.4 f 3. 4Hz