EE100 Lab 3 Experiment Guide: RC Circuits

Transcription

1 I. Inroducion EE100 Lab 3 Experimen Guide: A. apaciors A capacior is a passive elecronic componen ha sores energy in he form of an elecrosaic field. The uni of capaciance is he farad (coulomb/vol). Pracical capacior values usually lie in he picofarad (1 pf = F) o microfarad (1 µf = 10-6 F) range. Recall ha a curren is a flow of charges. When curren flows ino one plae of a capacior, he charges don' pass hrough (alhough o mainain local charge balance, an equal number of he same polariy charges leave he oher plae of he device) bu insead accumulae on ha plae, increasing he volage across he capacior. The volage across he capacior is direcly proporional o he charge Q sored on he plaes: Q = V (Eq. 2) Since Q is he inegraion of curren over ime, we can wrie: V = Q = i( ) d (Eq. 3) Differeniaing his equaion, we obain he I-V characerisic equaion for a capacior: dv i = (Eq. 4) d B. An R (resisor capacior) circui will have an exponenial volage response of he form v() = A B e -/R where consan A is he final volage and consan B is he difference beween he iniial and he final volages. (e x is e o he x power, where e = 2.718, he base of naural logarihms.) The produc R is called he ime consan (whose unis are seconds, if R is in ohms and is in farads), and is usually represened by he Greek leer τ. The volage can be wrien convenienly in erms of he iniial and final volages and he ime consan as v() = v(final) [v(iniial) v(final)]e -/τ When he ime has reached a value equal o he ime consan, τ, hen he volage is B e -τ/r = B e -1 = 0.37 * B vols away from he final value A, or abou 63% of he way from he iniial value (A B) o he final value (A). The characerisic exponenial decay associaed wih an R circui is imporan o undersand, because complicaed circuis can ofenimes be modeled simply as a resisor and a capacior. This is especially rue in inegraed circuis (Is). A simple R circui is drawn in Figure 1 wih currens and volages defined as shown. Equaion 5 is obained from Kirchhoff s Volage Law, which saes ha he algebraic sum of volage drops around a closed loop is zero. Equaion 6 is he defining I- V characerisic equaion for a capacior (as derived above), and Equaion 7 is he defining I-V characerisic equaion for a resisor (Ohm s Law). 1

2 VR _ VIN _ I V _ Figure 1 V = V V (Eq. 5) IN R I dv d = (Eq. 6) V R = IR (Eq. 7) By subsiuing Equaions 6 and 7 ino Equaion 5, he following firs-order linear differenial equaion is obained: dvc VIN = V R (Eq. 8) d If V IN is a sep funcion a ime = 0, hen V and V R are of he forms: V / R = A Be (Eq. 9) V R A B e / R = (Eq. 10) If a volage difference exiss across he resisor (i.e., V R <> 0), hen curren will flow (Eq. 7). This curren flows hrough he capacior and causes V o change (Eq. 6). V will increase (if I > 0) or decrease (if I < 0) exponenially wih ime, unil i reaches he value of V IN, a which ime he curren goes o zero (since V R = 0). For he square-wave funcion V IN shown in Figure 2a, he responses V and V R are shown in Figure 2b and Figure 2c, respecively. 2

3 Figure 2 (a) (b) (c) Noe ha if he frequency of he square wave V IN is oo high (i.e., if f>>1/r), hen V and V R will no have enough ime o reach heir asympoic values. If he frequency is oo low (i.e., if f<<1/r), he decay ime will be very shor relaive o he period of he waveform and hus he exponenial decay will be difficul o observe. As a rough guideline, he period of he square wave should be chosen such ha i is approximaely equal o 10R, in order for he responses shown in Figure 2b-c o be readily observed on an oscilloscope. II. Hands On A. Deermining he R ircui onfiguraion In his par of he experimen, you will make ohmmeer measuremens o see if you can discover a mehod o deermine if a resisor and capacior are conneced in series or in parallel. Figure 3 Figure 4 Ohmmeer V () Ohmmeer V () - - Series R ircui Parallel R ircui (a) Ge a resisor and capacior from your TA. Recall ha an ohmmeer has a buil-in curren source ha sends a small curren ino he circui under es. The ohmmeer reads he volage across he circui under es and deermines he resisance of he circui using Ohm s Law. 3

4 (b) Build he circui shown in Figure 3. Noe ha he ohmmeer s curren source keeps on charging up he capacior. (For small values of capaciance, he capacior will be fully charged almos insanly.) Quesion 1: Are you able o measure he value of he resisor? If no, explain he reason why you canno make he measuremen. (c) Build he circui shown in Figure 4. Noe ha he capacior sops charging when he curren hrough he resisor is equal o he curren from he ohmmeer. Quesion 2: Explain how you go your ohmmeer reading for he circui in Figure 4. Why does i ake some ime before he ohmmeer s reading sabilizes? Quesion 3: Given a black box wih eiher a series or parallel R circui, can you deermine he R configuraion using an ohmmeer? If so, how? B. Idenifying Physical Values in a Series R ircui Black Box and a Parallel R ircui Black Box The TA will give you wo black boxes (if available). One conains a series R circui and he oher conains a parallel R circui. Deermine he basic resisorcapacior configuraion in each black box using an ohmmeer. If you are insruced o build he black box yourself, noe ha you are no allowed access o he node shared by R B and B, so you can measure R B direcly. Please use he breadboard. Quesion 4: Deermine wheher each block box is a series or parallel R circui. 1) Series R ircui Black Box (a) onsruc he circui below for he black box ha conains he unknown resisor R B and capacior B in series. There is an R S = 50 ohms source resisor inside he funcion generaor (migh be negligible) and Rx is he exernal resisor ha is suggesed by he TA. Measure and record R X. 4

5 (b) Se he ampliude of he square wave o abou 5 V PP (no criical), wih 0 VD offse. Adjus he frequency of he square wave and he scope viewing scale unil you see he exponenial characerisic decay. (Make sure ha you are using he waveform from he OUTPUT erminal of he funcion generaor, no he SYN erminal.) (c) Noe ha he volage across Rx follows he shape of V IN V (ry he mah and look a he nice picures). V0 V IN () - V0 V0 V () V X () V X () V IN() - V () Figure 6: Volage waveforms in a series R circui (d) From volage V X across R X, you could use Ohm s Law o obain he curren I X, bu boh I X and V X will have he same ime consan. You should rigger on A1 edge and se Main Time Ref o Lef, and always use he knobs o zoom in upon your cursor measuremens. Your TA should review use of he URSOR funcion: Se cursor V1 o ground, se cursor V2 o he peak and record V(A1). Nex, move V2 o of ha value. Move 1 o he sar, move 2 o inersec V2. is he ime consan. 5

6 Vpo() ~ 250mV ~ 250mV V2 250mV x exp(-1) = 91.97mV V Time onsan τ1 = 2-1 Figure 7: How o measure he ime consan τ 1 on he oscilloscope Quesion 5: Wha is he ime consan τ 1? Wha is he value of R X? (d) Now increase he value of R X by abou a facor of en. Measure he new R X2. Quesion 6: Wha is he ime consan τ 2? Wha is he value of R X2? Quesion 7: Solve for he resisance R B and capaciance B using he formula for he ime consan in boh rials. τ = B (R S R B R X ). Ask your TA for he resisance and capaciance values. Are hey in good agreemen wih he values you have obained experimenally? Explain if here are any significan differences. 2) Parallel R ircui Black Box (a) Build he Parallel Black Box using unknown R B and B from your TA. Measure he resisance of he circui inside he black box using he ohmmeer. The res of he measuremens go oward finding he capaciance B. Quesion 8: Wha is he measured value of he resisor inside he black box? (b) Selec a resisor R X of a value comparable o he resisor R B and consruc he circui below. Measure he ime consan of he circui wih R X. 6

7 Figure 9: Seup for finding R and of an unknown parallel R circui Quesion 9: Measure he ime consan of he circui wih R X, hen solve for he value of he capacior B. Ask your TA for he values of he resisor and he capacior inside he black box. Are hey in good agreemen wih he values you have obained experimenally? Explain if here are any significan differences. 7