Lab 10: RC, RL, and RLC Circuis In his experimen, we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors. We will sudy he way volages and currens change in hese circuis when volages are suddenly applied or removed. To change he volage suddenly, a funcion generaor will be used. In order o observe hese rapid changes we will use an oscilloscope. 1. The Square Wave Generaor Inroducion We can quickly charge and discharge a capacior by using a funcion generaor se o generae a square wave. The oupu of his volage source is shown in Figure 1. 0 0 1 3 5 7 9 11 13 15 17 19 21 23 25 T Figure 1. Oupu of square-wave generaor One conrol on he generaor les you vary he ampliude, 0. You can change he ime period over which he cycle repeas iself, T, adjusing he repeiion frequency f = 1/T. The generaor is no an ideal volage source because i has an inernal resisance of 50 Ω. Thus, for purpose of analysis, he square-wave generaor may be replaced by he wo circuis shown in Figure 2. When he volage is on, he circui is a baery wih an EMF of 0 vols in series wih a 50-Ω resisor. When he volage is off, he circui is simply a 50-Ω resisor. R R 0 + On Off Figure 2. Square-wave generaor equivalan circui 1
Procedure To learn how o operae he oscilloscope and funcion generaor, se he funcion generaor for square wave oupu and connec he generaor o he verical inpu of he oscilloscope. Adjus he oscilloscope o obain each of he paerns shown in Figure 3. Try changing he ampliude and repeiion frequency of he generaor and observe wha corresponding changes are needed in he oscilloscope conrols o keep he race on he screen he same. Now se he funcion generaor o a frequency of abou 100 Hz. Observe paern and adjus he frequency unil he period T = 10.0 ms. Funcion Generaor Oscilloscope Figure 3. Observing he oupu of he square-wave generaor. 2. Resisance-Capaciance Circuis Inroducion We have previously sudied he behavior of capaciors and looked a he way capacior discharges hrough a resisor. Theory (see exbook) shows ha for a capacior, C, charging hough a resisor, R, he volage across he capacior,, varies wih ime according o where o is he final, equilibrium volage. () = o (1 e /RC ), (1) When he same capacior discharges hrough he same resisor, () = o e /RC (2) The produc of he resisance and capaciance, RC, governs he ime scale wih which he changes ake place. For his reason i is called he ime consan, which we call τ (au). I can be found indirecly by measuring he ime required for he volage o fall o o /2 (see Figure 4 below). This ime inerval is called he half-life, T 1/2, and is given by he equaion T 1/2 = (ln2)τ, so τ = T 1/2 /ln2 = T 1/2 /(0.693) (3) 2
0 10 8 1/2 0 6 4 2 0 0 2 4 6 8 10 12 14 18 20 T 1/2 Figure 4. Discharge of a capacior Procedure Assemble he circui shown in Figure 5. Wih iniial values R = 10 kω, C = 0.1 µf, and f = 100 Hz, observe one period of he charge and discharge of he capacior. Make sure he repeiion frequency is low enough so ha he volage across he capacior has ime o reach is final values, o and 0. Skech he waveform you observed on he daa shee. Use he ohmmeer o measure R. R = 10 kω C = 0.1 µf Funcion Generaor Oscilloscope Figure 5. Invesigaing an RC circui 3
Techniques Here is a mehod for finding T 1/2. Change oscilloscope gain (vols/cm) and sweep rae (ms/cm) unil you have a large paern on he screen. Make sure he sweep speed is in he calibraed posiion so he ime can be read off he x-axis. Cener he paern on he screen so ha he horizonal axis is in he cener of he paern. Tha is, so ha he waveform exends equal disances above and below he axis. Move he waveform o he righ unil he sar of he discharge of he capacior is on he verical axis (Figure 6b). You may find i helpful o expand, or magnify, he race. The sweep ime is now a facor of five or en faser han indicaed on he dial. Ask your insrucor for deails. The half-life, T 1/2 is jus he disance shown on Figure 6b. T 1/2 Figure 6a and b. Measuring he half-life 4
Procedure Measure he half-life, T 1/2 and from his compue he ime consan τ using Equaion 3. Compue he value of RC from componen values. Noe ha, as described above, he square-wave generaor has an inernal resisance of 50 Ω. Thus he oal resisance hrough which he RC circui charges and discharges is (R + 50 Ω). Try differen values of f and hence, T, while keeping τ fixed by no changing eiher R or C. Skech wha you saw when he period T of he square wave is much larger han he ime consan τ. Repea when i is much smaller. 3. Resisance-Inducance Circuis Inroducion In his par we conduc a similar sudy of a circui conaining a resisor and an inducor, L. Firs consider he circui shown in Figure 7, below. The ex shows ha if we sar wih he baery conneced o he LR circui, afer a long ime he curren reaches a seady-sae value, io = o /R. R 0 L Figure 7. A model circui wih an inducor and resisor If we call = 0 he ime when we suddenly hrow he swich o remove he baery, allowing curren o flow o ground. The curren changes wih ime according o he equaion i() = i o e (R/L) (4) If a a new = 0 we hrow he swich so he baery is conneced, he curren increases according o he equaion The ime consan for boh equaions is L/R and i() = i o (1 e (R/L) ) (5) L R = τ = T 1/2 0.693 (6) We can find he curren by measuring he volage across he resisor and using he relaionship i`= /R. 5
Noe ha wha we would firs see is he growh of curren given by Equaion 5, where he final curren depends on he square-wave ampliude o. Then, when he square wave drops o zero, he curren decays according o Equaion 4. The ime consan should be he same in boh cases. Procedure Se up he circui shown in Figure 8 below. Wih iniial values R= 1kΩ and L = 25mH, se he oscilloscope o view one period of exponenial growh and decay. Again, make sure ha f is low enough for he curren o reach is final values, io and 0. Sar wih f = 5 khz. Skech he paern. Measure he half-life. From his value, compue he ime consan τ. Measure he value of R and he dc resisance of he inducor wih an ohmmeer. Finally add he inernal resisance of he square-wave generaor o obain he oal resisance. Compue he value of L/R from he componens and compare wih τ found from he indirec measuremen above. Try differen values of T. In paricular, skech he waveform when τ is much larger han he period T of he square wave and when i is much smaller. L = 25 mh R = 1 kω Funcion Generaor Oscilloscope Figure 8. Invesigaing he LR circui 4. Resisance Inducance Capaciance circuis Inroducion As discussed in he exbook, a circui conaining an inducor and a capacior, an LC circui, is an elecrical analog o a simple harmonic oscillaor, consising of a block on a spring fasened o a rigid wall. L C k M Figure 1. LC Circui and is analog, a mechanical SHM Sysem In he same way ha, in he mechanical sysem, energy can be in he form of kineic energy of he block of mass M, or poenial energy of he spring wih 6
consan k, in he LC circui energy can reside in he magneic field of he inducor U = 1 2 Li2, or he capacior, U = 1 2 q2 /C. Boh he curren and he charge hen change in a sinusoidal manner. The frequency of he oscillaion is given by ω o = 1/ LC (7) All circuis have some resisance, and in he same way fricional forces damp mechanical SHM, resisance causes energy loss (i 2 R) which makes he charge decay in ime. where τ = 2L/R or q() = q o e /τ cos ω 1 (8) ω 1 = (ω o 2 1/τ 2 ) 1/2 (9) T 1/2 = ln2(2l/r) = (0.693)τ (10) For large τ he sysem is underdamped and he charge oscillaes, aking a long ime o reurn o zero. Noe ha in Equaion (3) when ω 2 o = 1/τ 2, ω 1, he argumen of he cosine funcion of Equaion (2), is zero a all imes. This condiion is called criical damping. This condiion exiss when R = 2 L/C When he resisor is larger han he criical value he sysem is overdamped. The charge acually akes longer o reurn o zero han in he criically damped case. Experimen The decaying oscillaions in he LRC circui can be observed using he same echnique as used o observe exponenial decay. Again, a square-wave generaor produces he same effec as a baery swiched on and off periodically. The scope measures he volage across C as a funcion of ime. 7
Procedure Assemble he circui of Figure 2. Use a small value of R, say, 47Ω. Be sure o reduce he signal generaor frequency o 100 Hz or below so you can see he enire damped oscillaion. Measure he period and calculae he frequency of he oscillaions. The period is NOT 0.01 s = 1/100 Hz, he repeiion frequency of he square wave. NOTE: You have acually measured ω 1, he damped frequency. This is slighly less han ω o, he undamped frequency of Equaion. 1 bu he difference is negligible. Compare he ω 1 you measured wih he calculaed ω o = 1/ LC. Noe ha in he equaions, R represens he sum of he resisance of he inducor, he inernal resisance of he square-wave generaor, 50 Ω, and he resisor you pu in. To sudy criical damping and overdamping, remove your fixed resisor and pu in is place a 5-kΩ variable resisor. Sar wih a small value of R. Skech he underdamped oscillaions. Increase R unil criical damping is reached; ha is, unil he oscillaions disappear. Skech his curve. Use he ohmmeer o measure he value of he variable (poeniomeer) resisor. Add he dc resisance of he inducor and 50Ω for he generaor and compare wih he prediced value for criical damping, R = 2 L/C. Wha happens when he resisance is larger han he criical-damping value? Skech your resuls. R L = 25 mh C = 0.1 µf Funcion Generaor Oscilloscope Figure 2. Invesigaing he LRC circui When he circui is underdamped, Equaion (2) applies. This means ha he ampliude of he oscillaion will decay exponenially, wih he ime consan for he decay being: τ = 2L/R. (11) Recall ha when an exponenial decay is ploed on a semi-log scale he resuling graph Is a sraigh line wih a slope equal o -1/τ. You can find he slope of a line on a semi-log graph by idenifying he wo end poins of he line. Noe he ime and volage a each poin 1 and 1, 2 and 2. 8
Calculae he naural log of he wo volages. Then (ln 2 - ln 1) /( 2-1 ) = -1/τ To measure he ime consan of he decay of he oscillaions, follow his procedure: Procedure Adjus he variable resisor so ha he circui Is underdamped and oscillaes abou seven or eigh imes before he oscillaions become oo small o be easily seen on he oscilloscope. Cener he oscillaion paern verically on he screen so ha when he oscillaions have decayed he line on he oscilloscope coincides wih he ime axis. Make a daa able recording he volage of each oscillaion peak, and he corresponding ime for each peak. When your able is complee you should have six or seven ses of daa recorded. Plo your daa on a semi-log graph. Following he procedure described above, deermine he ime consan for your circui. This is your experimenal value for he ime consan. Remove he variable resisor from he circui and measure is resisance. Add his value o he resisance of he square-wave generaor (50 ohms) and he resisance of he inducor o ge he oal resisance of your circui. Using Equaion (5), calculae he heoreical decay ime consan for your circui. Compare his heoreical value o he experimenal value you found above. They should agree wihin en or weny percen. If hey do no, consul your insrucor. 9
RC, RL, and RLC Circuis Name Parner Secion: 2. RC Circuis 1. Using dimensional analysis, show ha he produc of an ohm imes a farad (RC) has unis of ime. 2. Skech observed paern produced by he RC circui on op of he square wave. 3. Measured T 1/2 s τ= T 1/2 /0.693= s RC from componen values s Wihin he uncerainies of he olerances (10%) of he resisor and capacior, do your measuremens suppor he equaion τ = RC? If here is more han 20% disagreemen, see your insrucor. 4. Wha is he larges volage across he capacior (o)? Wha is he larges charge on he capacior (qo = C o) 10
5. Skeches. τ>> T τ = T = τ<< T τ = T = 6. How migh you measure he inernal resisance of your funcion generaor? 3. RL Circuis 1. Using dimensional analysis, show L/R has unis of ime. 2. Skech of across R. 3. MeasuredT 1/2 s; τ = T 1/2 /0.693 = s L/R from componen values s Wihin he uncerainies of he manufacuring olerances (10%) of he resisor and inducance, do your measuremens suppor he equaion τ = L/R? 4. Wha is he larges curren hrough he inducor (io)? 11
5. Skeches τ >> T τ = T = τ << T τ = T = I I 4 RLC Circuis 1. Measured period s. Calculaed f 1 = Hz; ω 1 = 2 π f 1 = rad/s. Compare wih he frequency calculaed from componen values, ω o = 1/ LC = rad/s. 2. Skech of paerns Underdamped Criically damped Overdamped 3. When criically damped, resisance of poeniomeer = Ω Sum of resisances of poeniomeer, inducor, and generaor = Ω Theoreical value R = 2 L/C = Ω. Comparison and discussion of any disagreemen: 4. Experimenal decay consan (from graph) = Theoreical decay consan (Equaion(11)) = Comparison and discussion of any disagreemen: 12