REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens: The Oscilloscope and Signal Generaor - Model 19 Sanley Wor and Richard F.M. Smih, Suden Reference Manual for Elecronic Insrumenaion Laboraories (Prenice Hall 1990). Also see he video - Physics Skills; How o use he Oscilloscope - sarring Dr D.M. Harrison. Circui Wiring: This Laboraory Manual: Circui- Wiring Techniques: Commonly Used Insrumens; he Oscilloscope. OBJECTIVES This lab will give you experience in: developing sraegies o handle complex equipmen such as oscilloscopes using an oscilloscope o measure a signal which varies regularly in ime planning and wiring a simple circui analyzing an exponenial funcion obained from daa readings using he daa analysis program on he Faraday compuer INTRODUCTION of oher comparable exponenial processes. There are numerous naural processes in which he rae of change of a quaniy is proporional o ha quaniy. One biological example is he case of populaion growh in which he rae of increase of he number of members of a species is proporional o he number presen. In his case he populaion is said o grow exponenially. A radioacive example is he case in which he rae of loss of he number of nuclei is proporional o he number of nuclei presen. The number of nuclei decreases exponenially. You migh hink
An elecrical example of exponenial decay is ha of he discharge of a capacior hrough a resisor. A capacior sores charge, and he volage V across he capacior is proporional o he charge q sored, given by he relaionship V = q/c, where C is called he capaciance. A resisor dissipaes elecrical energy, and he volage V across i is proporional o he curren (which is jus he rae of flow of dq charge) hrough i, given by V R #, where R is called d he resisance. When a charged capacior is conneced o a resisor, he charge flows ou of he capacior and he rae of loss of charge on he capacior as he charge flows hrough he resisor is proporional o he volage, and hus o he oal charge presen. This can be expressed as : R dq q dq so ha 1 (1) d C d RC q RC which has he exponenial soluion q q o e where q o is he iniial charge on he capacior (a ime = 0). As he volage across he capacior is proporional o is charge, he volage V displays RC he same exponenial behaviour ; divide boh sides by C o obain V V o e. This exponenial decay of volage may be likened o a waer ank wih a small hole in he boom, filled wih waer, in which he flow or rae of loss of volume of waer in he ank (analogous o he rae of loss of elecrical charge from he capacior) is proporional o he pressure of he waer as i exers iself on he hole (analogous o he volage across he resisor). As he Figure 2. pressure is proporional o he heigh of waer in he ank, which is in urn proporional o he oal volume of waer, we find ha he volume of waer and he pressure exhibi exponenial decay. In his experimen, insead of merely discharging an already charged capacior, you will be using an Alernaing Curren (AC) square wave volage supply o charge he capacior hrough he resisor many imes per second, firs in a posiivedirecion and hen in a negaive direcion. The charging process also exhibis he same exponenial behaviour as he discharge. However his ime he exponenial curve approaches a consan asympoic value raher han a zero value.
THE EXPONENTIAL The exponenial volage funcion, which is derived from equaion (1), - V() V (2) o e is shown in Figure 3. I has a slope (rae of change) which is proporional o he value of he funcion (V) no maer where you are on he curve. Noe ha, in equaion (2), when = -, V() falls o 1/e = 0.368 of is original value (a = 0). - is called he ime consan for he exponenial decay. The ime o drop o 1/e of a previous value is consan, no maer where on he curve you ake your "iniial" value. Figure 3 illusraes he exponenial decay for a discharging capacior, while Figure 4 illusraes he volage change for a charging capacior. In he laer case, he volage increases, bu sill approaches a consan value; his is sill exponenial decay, bu because he volage sars from a lower value and hen rises o is asympoic value, an addiional consan erm is needed in he analogue o Equaion (2). The full expression in his case is V() V o (1 e - ). In his experimen you will be observing repeaed exponenial curves; you can confirm wheher he decay in volage is exponenial, and measure he ime consan for ha decay.. Capacior Discharging Figure 3. Capacior Charging Figure 4.
THE EXPERIMENT Connec he signal generaor in series wih he resisor and capacior as shown in Figure 5. Noe: As wih all elecrical circuis, connec up he componens of he circui firs, hen inroduce he measuring equipmen (in his case he oscilloscope) only aferwards. Figure 5. Figure 6. Connecing he Y B and Y B channels of he oscilloscope as in Figure 5 will allow you o simulaneously observe he applied volage from he signal generaor (he square wave) wih beam A of he dual race oscilloscope, and he volage across he capacior, V C, wih beam B. The ground leads (black) of he coaxial cables, which should be conneced o he ground oupu of he signal generaor (also black), are denoed by G. The pairs of leads (G, Y A ) and (G, Y B ) represen he coaxial cables leading o he oscilloscope. Adjus he DC OFFSET of he generaor so ha he generaor oupu alernaes beween a posiive volage and zero volage. Use he manufacurer's values for C and R in your comparison of observaion and heory. You may noice ha he load placed on he signal generaor by he circui ends o disor he applied square wave; experimen wih he oupu volage of he signal generaor o minimize his effec. Repea your measuremens for a leas wo differen values for R C and hus for he ime consan. POINTS TO CONSIDER: The daa you ake should es wheher he volage across he discharging capacior V C shows exponenial behaviour Iniially choose values of frequency f which allow he capacior o charge or discharge fully in each period. (The period of he signal from he signal generaor T = 1/f should be several imes he ime consan -.) Try ou a variey of values of he signal generaor frequency and see wha
i does o your display. Obain a quick value for he ime consan -, by measuring, on he oscilloscope screen, he ime required for he volage o fall owards he asympoic value by a facor of 1/e. Use he oscilloscope o deermine ime and volage values for paricular values of R and C and record V C as a funcion of. If you use he daa creaion and analysis program on he Faraday compuer o analyze your resuls, noe ha he noaion used for ln(x), he naural logarihm of x, is Log[x]. Before dismanling he circui, you migh change he applied signal from a square wave o a sinusoidal volage; compare he applied volage o he volage across he capacior in his case. Make a qualiaive record of your observaions; can you give a qualiaive explanaion? COMMENTS ON THE OSCILLOSCOPE: One of he chief pieces of learning in his experimen is finding ou how o drive an oscilloscope. Thus, i is imporan ha you play wih he insrumen, learning wha he numerous conrols do by rying hem ou. The following commens may guide you o some of he conrol seings: The AC-0-DC swiches on he Y A and Y B secions should normally be se o DC. 0 is used if you wan o check wha he acual posiion of 0 vols is on he screen. AC alers he signal in a way ha is ofen very useful, bu i also disors he signal. Never use AC excep for special effecs ha you will learn abou in second year or if you ask your demonsraor. There are hree ses of rigger conrol buons. These ell he oscilloscope when (a wha poin) in he signal on he screen you wan he race o sar o be displayed. For he rigger source conrols, use Y A, or Y B, depending on wheher you wan he synchronizaion o ake place from he A or he B signal. For he rigger slope conrols (TRIG.) choose "+" or "", depending on wheher you wan he race on he screen o sar when i is rising (posiive slope) of falling (negaive slope). For he rigger mode conrols, choose eiher AUTO or AC, depending on which gives you he mos sable race. If you use AC mode, you may have o play wih he LEVEL conrol in order o ge a complee race. Noe ha he TIME/CM dial affecs he horizonal scaling only, and in no way moves your race up or down. Similarly, he wo Y A and Y B, AMPL dials affec he verical scales only, and in no way move he race sideways. When making quaniaive measuremens, make sure ha he small knobs on he op of he main TIME/CM and AMPL conrols are se o he CAL ( CALIBRATED ) posiion. (cp-1992,1993;k,jbv-1995)