Lagrangian for RLC circuits using analogy with the classical mechanics concepts

Transcription

1 Lagrangian for RLC circuis using analogy wih h classical mchanics concps Albrus Hariwangsa Panuluh and Asan Damanik Dparmn of Physics Educaion, Sanaa Dharma Univrsiy Kampus III USD Paingan, Maguwoharjo, Slman, Yogyakara, Indonsia panuluh@usd.ac.id Absrac. W sudy and formula h Lagrangian for h LC, RC, RL, and RLC circuis by using h analogy concp wih h mchanical problm in classical mchanics formulaions. W found ha h Lagrangian for h LC and RLC circuis ar govrnd by wo rms i.. kinic nrgy-lik and ponial nrgy-lik rms. Th Lagrangian for h RC circui is only a conribuion from h ponial nrgy-lik rm and h Lagrangian for h RL circui is only from h kinic nrgy-lik rm.. Inroducion Lagrangian formalism is a powrful way o obain h quaion of moion of a physical sysm. Th Lagrangian formalism is urnd up o solv problms ha ar no simpl by using Nwonian Mchanics []. In Nwonian mchanics, w usually formula h mchanical problm (physical sysm) in h form of forc or vcor. Manwhil, in Lagrangian mchanics, w approach h mchanical problm in h form of nrgy which is a scalar. If w know h Lagrangian of a physical sysm, hn w can obain h quaion of moion of a physical sysm by using h Eulr-Lagrang quaion []. In h lcrical subjc, h RLC (Rsisanc-Inducanc-Capacianc) circui is commonly discussd in sudn physics xbooks as an xampl of h applicaion of lcronics principls o build an lcronic dvic [3]. In h mahmaical physics subjc, h RLC circui is commonly usd o show h applicaion of diffrnial quaion in h physical sysm [4]. Th knowldg of RLC circui is crainly of gra physical inrs boh from xprimnal (applid) and horical sids. Rcnly, h RLC circui and is physical propris o b an inrsing rsarch subjc du o h dvlopmn of msoscopic physics and nanophysics ha nd a quanum mchanics bcaus h msoscopic sysm and nanophysics ar no anymor a macroscopic sysm. Th analysis of RLC circui as a msoscopic sysm by using quanum mchanics basd on Cardirola-Kanai Hamilonian and quanum invarian mhod o solv h Schrödingr quaion for h RLC circui and o obain h corrsponding wav funcions in rm of a paricular soluion of Miln- Pnny quaion was rpord by Pdrosa and Pinhiro [5]. Th quanum mchanics ramn of h RLC circui wih discr charg and smiclassical considraion can b found in Urras-Diaz [6] ha obaind h approximaion of nrgy ignvalus in rm of a dimnsionlss paramr L C h ( is h lcron charg, and h is h Planck consan). In liraur, w found mos analysis of h RC, RL, LC, and RLC circuis, as a msoscopic sysm or nanophysics, ar commonly discussd and formulad by using h concp of quanum mchanics [7-0]. As w hav alrady known ha h

2 nrgy opraor in quanum mchanics is h Hamilonian which is dfind as h sum of nrgy kinic and nrgy ponial of h physical sysm. In classical mchanics, w also know ha w can formula h physical sysm in form of Lagrangian dfind as h kinic nrgy minus ponial nrgy of h physical sysm. Thus, i is also an inrsing problm o know h xplici form of h Lagrangian for RC, LC, RL, and RLC circuis and is phnomnological implicaions. In his papr, w formula h Lagrangian for LC, RC, RL, and RLC circuis by using h analogy wih h classical mchanics formulaion for a physical sysm. W also discuss h advanags of h lagrangian formulaion of LC, RC, RL, and RLC circuis compar o h o lcrical formulaion as on can find in h mos of h sudn xbooks. W organizd his papr as follow: in scion w driv h Lagrangian for LC circui; in scion 3 w driv h Lagrangian formulaion for h RC circui; in scion 4, w formula h lagrangian for RL circui; in scion 5 w driv h Lagrangian for RLC circui. Finally, in scion 6 w wri h conclusions.. Lagrangian for h LC Circui In his scion, w formula h Lagrangian for h LC circui. W bgin wih by wriing h diffrnial quaion of Kirchhoff s rul for LC circui wihou sourc as follow d q q 0. () d LC By analogy wih h harmonic oscillaor, w can pu h angular frquncy rgarding L and C as follow. () LC As w know from h mahmaical physics, h gnral soluion of Eq. () is givn by i -i q( ) k k (3) whr w can drmin h consans k and k from h appropria boundary valus. Mahmaically, hr is wo possibl iniial condiions for h capacior C. Firs, a im = 0 h capacior has zro chargs: q(0) = 0 and a im = h charg of h capacior rach a maximum valu ha is Q and his procss is known as charging procss. Scond, a im = 0 i is possibl ha h capacior C has iniial charg Q 0 and a im = h capacior has no charg which is known as discharging a capacior. If w pu h h iniial condiion q(0) = 0 ino Eq. (3), hn w hav 0 0 q(0) 0 k k (4) ha implis k k (5) By insring Eq. (5) ino Eq. (3), w obain i i q( ) k( ). (6) If w us Eulr s quaion, hn Eq. (6) rads q( ) ik sin( ). (7) I is clar from Eq. (7) ha h charg as a sin funcion. Th soluion of h diffrnial quaion of Eq. () wih h iniial condiion q(0) = 0 is no rlvan o h physical raliy alhough i is possibl from h mahmaics poin of viw. Thrfor w discard h charging procss wih h soluion in h form of sin funcion. If w pu h iniial condiion q(0) Q0 k k for discharging procss ino Eq. (3), hn w hav k Q0 k. (8) By insring h Eq. (8) ino h Eq. (3) and afr doing a lil algbra, h Eq. (3) bcom -i q( ) Q0 iksin( ). (9) Th scond rm of Eq. (9) dos no saisfy h iniial condiion ha = q = 0, so w drop h scond rm of Eq. (9) and finally, w hav h soluion for discharging capacior in h LC circui as follow

3 -i q( ) Q0. (0) Now, w ar in h posiion o formula h Lagrangian for LC circui. To formula h lagrangian of a physical sysm, w mus know h ponial nrgy and kinic nrgy of h sysm. Th ponial nrgy for h LC circui is givn by V ( q) q. C () By analogy wih h ponial nrgy of harmonic oscillaor, w can pu h corrsponding paramrs as follow m L, x q, k. C () By insring Eq. (0) ino Eq. (), hn w find h ponial nrgy of h LC circui as follow i VLC Q0, C (3) and h kinic nrgy of oscillaor harmonic sysm is Tho mx. (4) By using h analogy in Eq. () and ak drivaiv of Eq. (0), w hav h kinic nrgy of h LC circui as follow i TLC L Q0. (5) From Eqs. (3) and (5), w finally can wri h Lagrangian for h LC circui as follow i i LLC L Q0 Q0, C (6) which is composd of wo rms ha can b pu as kinic nrgy-lik and ponial nrgy-lik. 3. Lagrangian for h RC Circui In his scion, w formula h Lagrangian for h RC circui. Th diffrnial quaion of Kirchhoff s rul for RC circui wihou a sourc is givn by dq q R 0. (7) d C I has alrady known ha hr ar also wo procsss ha can b assignd o h RC circui i.. charging or discharging of h capacior C. By ingraing Eq. (7), and using h valus of q = 0 a im = 0 and q = q a h im =, w hn hav / q ( ) RC. (8) From Eq. (8) i is apparn ha h soluion is unaccpabl bcaus if w pu, hn h charg bcoms zro which is conrary o h physical procss as a charging procss. By ingraing Eq. (8), and using h valus of q = Q 0 a im = 0 and q = q a im =, hn w obain / q( ) Q0 RC. (9) which is rlvan o h physical procss as prviously sad. I is clar ha hr is no kinic nrgy rm for h RC circui. Th ponial nrgy in h RC circui as follows / RC VRC Q0. (0) C Bcaus hr is no kinic nrgy in h RC circui, hn h Lagrangian of RC circui as follows / RC LRC Q0. () C

4 4. Lagrangian for h RL Circui Th diffrnial quaion for h RL circui is givn by di L RI 0 d. () Th soluion of h Eq. () rad / I( ) I0 R L. (3) For h RL circui, w can mak an analogy wih h quaion of moion of a paricl wih h forc is vlociy dpndn as follow dv m kv 0, (4) d By puing L corrspond o mass m and h rsisanc R corrspond o fricion consan k, hn w can pu h currn I in Eq. (3) as a corrsponding vlociy in Eq. (4). To formula h kinic nrgy of h RL circui, w us h analogy concp by using h usual mchanical kinic nrgy ½ mv and lcric currn I corrspond wih vlociy v, and hn w hav R / L TRL LI0 (5) as h kinic nrgy rm for RL circui. For h RL circui, hr is no ponial nrgy rm. Thus, h lagrangian for h RL circui rad R / L LRL LI0. (6) 5. Lagrangian for h RLC Circui Th diffrnial quaion for h RLC circui as follows d q dq q L R 0 d d C or d q R dq q 0 (7) d L d LC By using analogy wih a dampd harmonic oscillaor, w hav R, 0 (8) L LC whr γ and ω 0 as damping facor and naural frquncy rspcivly. Th gnral soluion of Eq. (7) as follows ( ) ( ) q( ) C C (9) whr 0. (30) Thr ar hr possibl cass for h paramr i.. ovrdamping, criical damping and undrdamping. 5. Ovrdamping Ovrdamping will occur whn λ > 0 so h xponns in h firs and scond rm in Eq. (9) ar ral. If w insr h discharging capacior boundary condiion ino Eq. (9), hn w hav h soluion as follow ( ) q( ) Q0. (3) By using h sam procdur o formula h Lagrangian for LC circui, w hn hav h Lagrangian for ovrdamping RLC circui as follow

5 ( ) ( ) 0 ( ) 0 Q LRLCovr L Q. (3) C Afr doing a lil algbra ha is by xpanding and simplifying λ and γ, h Eq. (3) bcom R R 4 R R 4 L L LC R R R 4 L L LC Q0 Q 0 L C L LC C LRLCovr. (33) On can s ha h Lagrangian for ovrdamping RLC circui as shown in Eq. (33) is composd of wo rms i.. kinic nrgy-lik and ponial nrgy-lik. 5. Criical Damping Criical damping will occur whn λ = 0. As shown in [6], h Eq (9) bcoms q( ) C C. (34) By insring h discharging capacior boundary condiion hn Eq (34) bcoms q( ) Q0 (35) and by using h sam procdur as prviously, w find h Lagrangian for h criical damping RLC circui as follow R / L R R / L Q0 LRLCcri Q0, (36) 8 L C which is composd of wo rms i.. nrgy kinic lik and ponial nrgy-lik. 5.3 Undrdamping Th RLC circui will undrgo h undrdamping whn λ < 0, and h gnral soluion for h cas of undrdamping RLC circui can b rad in [6] q( ) Acos( d 0) (37) whr R 0. (38) LC 4L By insring h discharging capacior boundary condiion and w s h iniial phas angl θ 0 by 0 o, hn Eq (37) bcom q( ) Q0 cos( ) (39) and w obain h lagrangian for h undrdamping RLC circui as follow R /L R Q0 cos 0 cos 0 sin R LRLCund L Q Q (40) L LC 4L C By inspcing Eq. (4), w can s ha h firs rm is kinic nrgy-lik and h scond rm is ponial nrgy-lik. I is clar from Eqs. (33), (36), and (40) ha h Lagrangians for h RLC circui ar h sum of h kinic nrgy-lik and ponial nrgy-lik as wll as on can find in classical mchanics. I can b sad ha h Lagrangian rms of h RLC circui indpndn of h valu of λ qualiaivly. 6. Conclusions W hav sudid and valuad h Lagrangian for LC, RC, LR, and RLC circuis by using h analogy wih h classical mchanics concp o formula h lagrangian of a physical sysm. W find ha h lagrangian for LC and RLC circuis ar composd of rms ha can b assignd as kinic nrgy and ponial nrgy rms in corrsponding wih h Lagrangian of a physical sysm in classical mchanics. Manwhil, h Lagrangian for h RC circui is only from ponial nrgy-lik conribuion and h and h lagrangian for h LR circui is only kinic nrgy-lik conribuion. W

6 also find ha h conribuion rms o h Lagrangian of h LRC circui indpndn of h valu of λ qualiaivly. Rfrncs [] Arya A P 998 Inroducion o Classical Mchanics nd Ediion (Nw Jrsy: Prnic-Hall) [] Fowls R and Cassiday G L 005 Analyical Mchanics 7h Ediion (Conncicu: Thomson Larning Inc.) [3] Srway R A and Jw J W 008 Physics for Sciniss and Enginrs wih Modrn Physics 7h Ediion (Conncicu: Thomson Larning Inc.) [4] Boas M L 006 Mahmaical Mhods in h Physical Scincs 3rd Ediion (Nw Jrsy: John Wily&Sons) [5] Pdrosa I A and Pinhiro A P 0 On h quanizaion of h dissipaiv sysms Prog. Thor. Phys [6] Urras-Diaz C A 008 Discr charg quanum circuis and lcrical rsianc Phys. L. A [7] Buo F A 993 Msoscopic physics and nanolcronics: nanoscinc and nanochnology Phys. Rp [8] Li Y Q and Chn B 996 Quanum hory for msoscopic lcric circuis Phys. Rv. B [9] Lu T and Li Y Q 00 Msoscopic circui wih linar dissipaion Mod. Phys. L. B [0] Flors J C and Urraz-Diaz C A 00 Msoscopic circuis wih charg discrnss: Quanum currn magnificaion for muual inducancs Phys. Rv. B