TWO-ELEMENT DC-DRIVEN SERIES LRC CIRCUITS

Transcription

1 TWO-ELEMENT D-DRIVEN SERIES LR IRUITS TWO-ELEMENT D-DRIVEN SERIES LR IRUITS by K. Franlin, P. Signell, and J. Kovacs Michigan Sae Universiy 1. Inroducion Relaing Q() and Is Derivaives o L, R, Mechanical Analogs a. The Mechanical Series ircui b. Mechanical-Elecrical orrespondences The L ase a. The L ircui b. Saring an Oscillaion c. Oscillaions d. Mahemaical Soluion by Analogy The RL and R ases Acnowledgmens Projec PHYSNET Physics Bldg. Michigan Sae Universiy Eas Lansing, MI 1

2 ID Shee: Tile: Two-Elemen D-Driven Series LR ircuis Auhor: K. Franlin and P. Signell and J. Kovacs, Michigan Sae Universiy Version: 2/1/2000 Evaluaion: Sage 0 Lengh: 1 hr; 16 pages Inpu Sills: 1. Describe he moion of a damped harmonic oscillaor (MISN-0-29). 2. Sae he node and juncion circui rules (MISN-0-119). 3. Sae he relaionship beween volage and charge in a capacior (MISN-0-135). 4. Sae he relaionship beween volage and curren in an inducor (MISN-0-144). Oupu Sills (Knowledge): K1. Sar wih he relaions for he poenial drops across each of he hree ypes of passive circui elemens and derive he relaion beween he curren and he imporan circui parameers for any wo-elemen series D-driven LR circui. K2. Lis he mechanical analogs of he circui componens and imporan circui parameers for wo-elemen D-driven LR circuis. K3. Given any wo-elemen D-driven LR circui, use analogies wih he damped harmonic oscillaor o sech a graph of he ime dependence of he charge on he capacior or he curren in he circui. Pos-Opions: 1. Three-Elemen D..-Driven Series LR ircui (MISN-0-152). 2. Resonances in LR ircuis (MISN-0-154). THIS IS A DEVELOPMENTAL-STAGE PUBLIATION OF PROJET PHYSNET The goal of our projec is o assis a newor of educaors and scieniss in ransferring physics from one person o anoher. We suppor manuscrip processing and disribuion, along wih communicaion and informaion sysems. We also wor wih employers o idenify basic scienific sills as well as physics opics ha are needed in science and echnology. A number of our publicaions are aimed a assising users in acquiring such sills. Our publicaions are designed: (i) o be updaed quicly in response o field ess and new scienific developmens; (ii) o be used in boh classroom and professional seings; (iii) o show he prerequisie dependencies exising among he various chuns of physics nowledge and sill, as a guide boh o menal organizaion and o use of he maerials; and (iv) o be adaped quicly o specific user needs ranging from single-sill insrucion o complee cusom exboos. New auhors, reviewers and field esers are welcome. PROJET STAFF Andrew Schnepp Eugene Kales Peer Signell Webmaser Graphics Projec Direcor ADVISORY OMMITTEE D. Alan Bromley Yale Universiy E. Leonard Jossem The Ohio Sae Universiy A. A. Srassenburg S. U. N. Y., Sony Broo Views expressed in a module are hose of he module auhor(s) and are no necessarily hose of oher projec paricipans. c 2001, Peer Signell for Projec PHYSNET, Physics-Asronomy Bldg., Mich. Sae Univ., E. Lansing, MI 48824; (517) For our liberal use policies see: hp:// 3 4

3 1 2 TWO-ELEMENT D-DRIVEN SERIES LR IRUITS by K. Franlin, P. Signell, and J. Kovacs Michigan Sae Universiy x spring (siffness ) m mass 1. Inroducion Suppose we now how o analyze a circui conaining jus capaciors, or jus resisors, or jus self-inducances. Wha happens when wo of hese hree componens are conneced in series? In he presen uni we derive he basic equaions governing such circuis and describe he use of mechanical analogs in solving hem. In anoher uni we rea he case where all hree ypes of circui elemens are presen, woring oward he general analysis and synhesis of circuis. 2. Relaing Q() and Is Derivaives o L, R, The circui shown in Fig. 1 is a closed loop. Thus: V 0 V R V L = V, (1) where V R sands for he poenial drop across he resisor, and similarly for he capacior and inducor. We can apply Ohm s Law o he resisor and he corresponding relaions for V and V L. 1 Subsiuing hese ino Eq. (1) gives us: V 0 = IR + L I + Q, (2) where dos sand for derivaives wih respec o ime. The signs show ha all hree elemens are assumed o oppose he applied volage V 0, meaning V 0 1 See MISN and MISN L R Figure 1. Swich is open for < 0 Swich is closed for > 0. shoc absorber (damping ) fricionless Figure 2. Mechanical Analog for he series LR ircui. ha he volage source mus do wor o pu charge on he capacior, o pu curren hrough he resisor, and o change he curren going hrough he inducor. Using he definiion of curren, I = dq/d, we conver Eq. (2) o a relaion for he charge on he capacior: V 0 = L Q + R Q + 1 Q. (3) 3. Mechanical Analogs 3a. The Mechanical Series ircui. ompare Eq. (3) o he force equaion for a damped driven spring 2 (see Fig. 2): = mẍ + λẋ + x, (4) where is he spring consan (a measure of is siffness), m is is mass, and λ is he circui s damping coefficien (his can include damping in he spring iself or may be mainly due o a damping device such as a shoc absorber or a dashpo). 3b. Mechanical-Elecrical orrespondences. Noice he similariy beween Eqs. (3) and (4). The correspondences are lised in Table 1. 2 See The Damped Driven Oscillaor, MISN

4 3 Table 1: Analogies Beween he LR ircui and he Damped Driven Oscillaor. LR ircui Oscillaor Inducor L Mass m Resisor R Shoc Absorber λ apacior 1 Spring Applied Volage V 0 Applied Force harge Q Posiion x urren Q = I Velociy ẋ = v Noice in Table 1 ha a large capacior is he analog of a wea spring. Tha is because, for a given amoun of charge sored in i, a larger capacior has less poenial difference across i han does a smaller capacior. Pu anoher way, i is easier o srech a wea spring o a larger displacemen jus as i is easier o pu more charge ino a large capaciance. 4. The L ase 4a. The L ircui. We begin he examinaion of wo-elemen circuis wih he L case, illusraed in Fig. 3. This case is especially ineresing because i is lie he perfec spring aached o a mass. Tha means here is no fricion, no energy dissipaion. In he elecric circui case i means no resisance so no energy dissipaion. In eiher sysem, mechanical or elecrical, an oscillaion once sared will go on forever. Of course in real life such sysems are no possible, bu one can come prey close. 4b. Saring an Oscillaion. In he mechanical case one sars an oscillaion be applying a force o he mass so as o srech he spring away from zero displacemen from equilibrium. One hen les go of he mass, perhaps by opening one s fingers, and oscillaions begin. In he elecrical case one usually isolaes he capacior and charges i from a 4 x() x 0 2 m l Q 0 Q() 2 L Figure 4. omparison of soluions in he illusraive example. baery. One hen disconnecs he baery and connecs he capacior o he oher inducor and he oscillaions begin. 4c. Oscillaions. In he mechanical oscillaions case, he displacemen is a sinusoidal funcion of ime, alernaely aing on posiive and negaive values wih respec o he equilibrium posiion. In he elecrical oscillaions case he charge on he capacior is a sinusoidal funcion of ime, alernaely aing on posiive and negaive values. In he mechanical oscillaions case he energy alernaes beween ineic energy sored in he moion of he mass and poenial energy sored in he compression or exension of he spring. In he elecrical oscillaions case he energy alernaes beween being sored in he inducor s magneic field and in he capacior s elecric field. 4d. Mahemaical Soluion by Analogy. onsider he circui shown in Fig. 3. Suppose he capacior is charged by an exernal source, which is hen removed a ime = 0. Wha happens o he capacior charge as ime progresses ( closed)? The equivalen mechanical model is jus ha of Fig. 2 afer removal of he shoc-absorber and he applied force (R = 0, V 0 = 0). This leaves a mass wih a spring aached, where he mass is displaced from equilibrium and hen released. The resul is simple harmonic moion: x() = x 0 sin ω, where ω = /m. Using our analogies in Table 1, L Figure 3. A wo-elemen L circui Q() = Q 0 sin ω, where ω = 1/L. These wo equaions are graphed in Fig

5 5 PS-1 The period of oscillaion for he L circui is P = 2π/ω = 2π L. 5. The RL and R ases PROBLEM SUPPLEMENT 1. The circui for his problem is shown in he sech below. Wih hree available elemens, L, R, and, here are hree possible wo-elemen circuis: L, RL, and R. The L case was reaed in Sec. 4; he oher wo are covered in his module s Problem Supplemen. a. V 0 L Acnowledgmens Preparaion of his module was suppored in par by he Naional Science Foundaion, Division of Science Educaion Developmen and Research, hrough Gran #SED o Michigan Sae Universiy. Draw he mechanical analog for he circui, omiing hose pars of he general model shown in Fig. 2 ha do no have corresponding elemens in he circui. b. In he mechanical case, he applied force does no affec he period or he ampliude of he oscillaion of he mass; i merely shifs he equilibrium poin abou which he oscillaion occurs. The ime dependence of x is hus idenical o he equaion found in he example, excep for an added consan and a phase shif: x() = x 1 + x 0 sin(ω + φ), where x 1 = /, and x 0 and φ depend on iniial condiions. Using he appropriae analogies, wrie he equaion for he ime dependence of Q in he above L circui. c. Given q(0) = 0 and I(0) = 0 for he above circui, can you evaluae Q 0 and/or φ in your answer o par (b) above? If so, wha are hey? d. Graph Q() vs. for his circui. e. By aing he derivaive of he charge funcion in boh he illusraive example and your answer o par (b), find he shif in curren caused by he inroducion of he induced volage in he above circui. Assume Q(0) = 0 in boh cases. 2. The circui for his problem is shown in he sech below. a. V 0 R L Draw he mechanical analog for he circui, omiing hose pars of he general model shown in Fig. 2 ha do no have corresponding elemens in he circui. 9 10

6 PS-2 PS-3 b. The shoc absorber in he mechanical case resiss any acceleraion by exering a force proporional o is rae of conracion. The proporionaliy consan is λ. Thus when he force is applied o he mass, i acceleraes unil he resisance force jus equals he applied force. The compression hen coninues a a consan rae. In he elecrical case, he induced volage across he inducor causes an increase in curren, which causes an increase in he poenial difference across he resisor. As his difference approaches V 0, he ne curren change per uni ime goes o zero, and he curren approaches a consan value. The equaion for he velociy of he mass in he mechanical case is: c. v() = λ ( 1 e λ/m). Wrie he corresponding equaion for he elecrical case. F 0 v() I() Here are graphs of he wo equaions of par (b). Noice ha velociy and curren ime are graphed, since here is no accumulaing charge. Wha analogy can be made o he posiion of he mass in he mechanical model? V R 0 3. The circui for his problem is shown in he sech below. a. V 0 Draw he mechanical analog for he circui, omiing hose pars of he general model shown in Fig. 2 ha do no have corresponding elemens in he circui. b. The force equaion for he mechanical case is: R = λẋ + x. There is no mass, so he sysem will jus shif unil i is in saic equilibrium. The shoc absorber and spring wor ogeher o preven he overshoo effec discussed in he illusraive example. This effec can be seen more easily by sudying a graph of he soluion, which is: Graphed, his is: x() x() = ( 1 e /λ). Here shoc absorber resisance plus increasing spring force reduce velociy o almos zero. Here spring compresses easily velociy is high. Noe ha on he lef side of he curve he spring compresses easily and he velociy is high. On he righ side of he curve he shoc absorber resisance plus he increasing spring force reduce he velociy almos o zero. Now, using our analogy, wrie he ime dependence of charge for he above circui. c. Sech a graph of Q() vs. for his circui. Brief Answers: 1. a. m fricionless b. Q() = Q 1 + Q 0 sin(ω + φ), where Q 1 = V 0 and ω = 1/L

7 PS-4 ME-1 c. φ = π/2. Q 0 mus be found using a second iniial condiion. d. Q() V 0 MODEL EXAM 1. see Oupu Sills K1-K3 in his module s ID Shee. P = 2 L Brief Answers: e. Zero. 2. a. 1. See his module s ex. m fricionless 3. a. b. I() = (V 0 /R) ( 1 e R/L). c. I = Q (he definiion of curren). So if we inegrae I, a any poin in he circui, from 0 o, we will ge he ne movemen of charge in he circui, Q. b. Q() = V 0 ( 1 e /R ). c. V 0 Q() 13 14

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