General Article Application of differential equation in L-R and C-R circuit analysis by classical method. Abstract
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1 Applicaion of Diffrnial... Gnral Aricl Applicaion of diffrnial uaion in - and C- circui analysis by classical mhod. ajndra Prasad gmi curr, Dparmn of Mahmaics, P.N. Campus, Pokhara rajndraprasadrgmi@yahoo.com Absrac Th papr dals wih h analysis of - and C- circui by using linar diffrnial uaion of firs ordr. A circui conaining an inducanc or a capacior C and rsisor wih currn and volag variabl givn by diffrnial uaion. Th gnral soluion of diffrnial uaion hav wo pars complmnary funcion C.F and paricular ingralp.i in which C.F. rprsn ransin rspons and P.I. rprsn sady rspons. Th gnral soluion of diffrnial uaion rprsn h compl rspons of nwork.in his conncion, his papr includs -, C- circui and ordinary diffrnial uaion of firs ordr and is soluion,. Ky words: circui analysis, classical mhod, - and C- circui, ordinary diffrnial uaion,,. Inroducion An uaion which involvs diffrnial cofficin is calld diffrnial uaion. A diffrnial uaion involving drivaivs wih rspc o singl indpndn variabl is calld ordinary diffrnial uaion and involving parial drivaivs wih rspc o mor han on indpndn variabl is calld parial diffrnial uaion. Th inr-conncion of simpl lcric dvic in which hr is a las on closd pah for currn o flow is calld lcric circui. Th circui is swich from on condiion o anohr by chang in h applid sourc or a chang in h circui lmns hr is a ransiion priod during which h branch currn and volag changs from hir formr valus o nw ons. This priod is calld ransin. Afr h ransin has passd h circui is said o b sady sa. Th linar diffrnial 4
2 Janapriya Journal of Inrdsciplinary Sudis, ol. 5 Dcmbr 26 uaion ha dscribs h circui will hav wo pars o is soluion h complmnary funcion corrsponds o h ransin and h paricular soluion corrsponds o sady sa. Th v-i rlaion for an inducor or capacior is a diffrnial. A circui conaining an inducanc or a capacior C and rsisor wih currn and volag variabl givn by diffrnial uaion of h sam form. I is a linar firs ordr diffrnial uaion wih consan cofficin whn h valu of,,c ar consan. and C ar sorag lmns. Circui has wo sorag lmns lik on and on C ar rfrrd o as scond ordr circui. Thrfor, h sris or paralll combinaion of and or and C ar firs ordr circui and C in sris or paralll ar scond ordr circui. Th circui changs ar assumd o occur a im and rprsnd by a swich. Th swich may b supposd o closd on and opn off a. Th ordr of diffrnial uaion rprsn drivaivs involv and is ual o h numbr of nrgy soring lmns and diffrnial uaion considrd as ordinary. h diffrnial uaion ha formd for ransin analysis will b linar ordinary diffrnial uaion wih consan cofficin. Th valu of volag and currn during h ransin priod ar known as ransin rspons. Th C.F. of diffrnial uaion rprsns h ransin rspons. Th valu of volag and currn afr h ransin has did ou ar known as sady sa rspons. Th P.I. of diffrnial uaion rprsns h sady sa rspons. Th compl or oal rspons of nwork is h sum of h ransin rspons and sady sa rspons which is rprsnd by gnral soluion of diffrnial uaion. Th valu of volag and currn ha rsul from iniial condiions whn inpu funcion is zro ar calld zro inpu rspons. Th valu of volag and currn for h inpu funcion which is applid whn all iniial condiion ar zro calld zro sa rspons. 5
3 Applicaion of Diffrnial... Tabl Elmns symbol and unis of masurmns S.No. lmn symbol uni charg Coulomb 2 currn i Ampr 3 rsisanc Ohm 4 inducanc Hnry 5 Capacianc C Farad 6 volag vol Daa and Mhods Th papr uss scondary sourcs and abl whr ncssary. Th publishd journal and books rlad o diffrnial uaion, circui and sysms mahmaical physics and lcrical nginring and lcriciy from various publishrs ar h scondary sourcs as indicad in rfrnc scion. suls and Discussion To sudy h ransins and sady sa in lcric circui, i is ncssary o know h mahmaical concp of diffrnial uaion and is soluion by classical mhod. 6
4 Janapriya Journal of Inrdsciplinary Sudis, ol. 5 Dcmbr 26 Firs ordr homognous diffrnial uaion. dy + py d dy y pd Ingraing, lny-p+lnk y k p Firs ordr non homognous diffrnial uaion dy + py Q d Th uaion is no alrd by muliplying p p d dy p + py d p { y } d Q p p Q d + k y. p Q p d + k y. p Q p d + k p Th firs rm of abov soluion is known as paricular Ingral and scond is known as complmnary funcion. Paricular Ingral dos no conains any arbirary consan and C.F. dos no dpnd on h forcing funcion Q. If Q is consan. Thn 7
5 Applicaion of Diffrnial... y. p p Q p + k p y. Q p + k p Th formaion of diffrnial uaion for an lcric circui dpnds upon h following laws. d i i d ii olag drop across rsisanc i di iii olag drop across inducanc d iv olag drop across capacianc C c Kirchhoff s law: h algbraic sum of h volag drop around any closd circui is ual o rsulan mf in h circui. Currn law: a a juncion currn coming is ual o currn going. di di E - sris circui: i+ d E d + i d C- sris circui: i+ c E d + c E - circui analysis 8
6 Janapriya Journal of Inrdsciplinary Sudis, ol. 5 Dcmbr 26 Th swich s is closd a im. Find h currn i hrough h volag across h rsisr and inducor. Hr, h volag across rsisanc i di olag drop across inducanc d di + i From Kirchhoff s law, d di + i d Which is firs ordr linar diffrnial uaion. d I.F Gnral soluion is, i d + k. + k. + k 9
7 2 k i +... Sinc h inducor bhavs as a opn circui. + i k k from +.. i Th volag across h rsisor and inducor ar givn as i. i d di.. A, i and, A, i and, A τ, i.632 and Applicaion of Diffrnial...
8 Janapriya Journal of Inrdsciplinary Sudis, ol. 5 Dcmbr , 632 τ is known as h im consan of h circui and is dfind as h inrval afr which currn or volag changs 63.2 prcn of is oal chang. C- circui analysis A condnsr of capaciy C farads wih is dischargd hrough a rsisanc ohms. Show ha if coulomb is h charg on h condnsr, i ampr h d currn and h volag a im, c, i and i. Thn C d Hr, h volag across rsisanci olag drop across capacianc C di + i From krichhoff s law, d di + i d 2
9 Applicaion of Diffrnial... Whn afr rlas of ky h condnsr gs dischargd and a ha im volag across h bary gs zro. So. d Th diffrnial uaion of abov circui is i + + C d c d d c d d c Ingraing, log + A c Bu a, h charg a condnsr is. Thrfor log A From log + log c c c c c c c Conclusion By using firs ordr ordinary diffrnial uaion in - and C- circui w can find h currni and volag v in h circui whn inducanc or capaciancc and rsisanc ar givn. frncs Das,H.K.& rma,..2. Highr Enginring Mahmaics. Nw Dlhi: S. Chand and company TD. Das, B.C,& Mukhrj, B.N 98. Ingral Calculus and diffrnial uaion.calcua:u.n.dhur and Sons Priva TD. 22
10 Janapriya Journal of Inrdsciplinary Sudis, ol. 5 Dcmbr 26 Gupa,S.2. A cours in Elcrical circui Analysis. Nw Dlhi: Dhanpa ai Publicaion Priva imid. Harpr,C.999. Inroducion o Mahmaical Physics. Nw Dlhi: Prnic Hall of India Priva imid. Kryszig,E.997. Advanc Enginring Mahmaics. Nw Dlhi: Nw Ag Inrnaional Priva imid. Mishra, P,uchi,..P. & Mohi Advanc Enginring Mahmaics. Nw Dlhi:.P.Mishra publicaion. Pan, G.D& Shrsha, G.S 266. Ingral Calculus and diffrnial uaion.kahmandu:sunila Prakashan. Sharma,.D., Poudl,T. N.& Adhakari, H. P Enginring Mahmaics II. Kahmandu :Sukanda Pusak Bhawan. Shrsha,.K.K, gmi,. P., Poudl,M. P. & Pandy, H..25. Fundamnal of Enginring Mahmaics olum II. Pokhara:Ozon Books and Saionr. Soni,.K.M.23.Circuis and Sysm. Dlhi: S.K.Kaaria and Sons. Tiwari,S.N.& Saroor, A.S.992.A firs cours in Elcrical Enginring. Allahabad: Whlr Publicaion. alknburg,.2.nwork Analysis.Nw Dlhi: Prnic Hall of India Priva imid. 23
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