8.02Experiment 4: RC CircuitsINTRODUCTIONRC CircuitsFigure 1 (a) RC circuit (b) Circuit diagram for t > 0In this lab you will be faced with an exponentially decaying current I = I0 exp(-t/) and an exponentially decaying voltage V = V0 exp(-t/). From data of voltage vs. time (or current vs. time)s, you will want to extract the time constant . We will do this in two different ways, using the “two-point method” or the “logarithmic method,” depicted in Fig. 7.In the two-point method (Fig. 7a) we choose two points on the curve (t1,I1) and (t2, I2). Because the current obeys an exponential decay, I = I0 exp(-t/), we can extract the time constant most easily by picking I2 such that I2 = I1/e. We should, in theory, be able to find this for any t1, as long as we don’t switch the battery off (or on) before enough time has passed. In practice the current will eventually get low enough that we won’t be able to accurately measure it. Having made this selection, = t2 – t1.2. AC/DC Electronics Lab Circuit BoardGENERALIZED PROCEDUREThis lab consists of two main parts. In each you will set up a circuit and measure voltage and current while the battery periodically turns on and off.In this part you will create a series RC (resistor/capacitor) circuit with the battery turning on and off so that the capacitor charges then discharges. You will measure the time constant using both methods described above and use this measurement to determine the capacitance of the capacitor.In this part you will add a second resistor in parallel with the capacitor to confirm your understanding of the in class problem worked before this part of the lab.END OF PRE-LAB READINGIN-LAB ACTIVITIESEXPERIMENTAL SETUPMEASUREMENTSPart 1: Measuring Voltage and Current in an RC CircuitPart 2: Measuring Voltage and Current in a parallel RC CircuitMASSACHUSETTS INSTITUTE OF TECHNOLOGYDepartment of Physics8.02Experiment 4: RC CircuitsOBJECTIVES 1. To explore the time dependent behavior of RC Circuits2. To understand how to measure the time constant of such circuitsPRE-LAB READINGINTRODUCTIONIn this lab we will continue our investigation of DC circuits, now including, along withour “battery” and resistors, capacitors (RC circuits). We will measure the relationshipbetween current and voltage in a capacitor, and study the time dependent behavior of RCcircuits.The Details: CapacitorsCapacitors store charge, and develop a voltage drop V across them proportional to theamount of charge Q that they have stored: V = Q/C. The constant of proportionality C isthe capacitance (in Farads = Coulombs/Volt), and determines how easily the capacitorcan store charge. Typical circuit capacitors range from picofarads (1 pF = 10-12 F) tomillifarads (1 mF = 10-3 F). In this lab we will use microfarad capacitors (1 F = 10-6 F).RC CircuitsConsider the circuit shown in Figure 1. The capacitor (initially uncharged) is connectedto a voltage source of constant emf E. At t = 0, the switch S is closed.Figure 1 (a) RC circuit (b) Circuit diagram for t > 0E04-1(a)(b)In class we derived expressions for the time-dependent charge on, voltage across, andcurrent through the capacitor, but even without solving differential equations a littlethought should allow us to get a good idea of what happens. Initially the capacitor isuncharged and hence has no voltage drop across it (it acts like a wire or “short circuit”).This means that the full voltage rise of the battery is dropped across the resistor, andhence current must be flowing in the circuit (VR = IR). As time goes on, this current will“charge up” the capacitor – the charge on it and the voltage drop across it will increase,and hence the voltage drop across the resistor and the current in the circuit will decrease.This idea is captured in the graphs of Fig. 2.Vf=εQf=CεQCapacitor, VCapacitorTimeVR,0=εI0=ε/RVResistor, ITimeFigure 2 (a) Voltage across and charge on the capacitor increase as a function of time while (b) the voltage across the resistor and hence current in the circuit decrease.After the capacitor is “fully charged,” with its voltage essentially equal to the voltage of thebattery, the capacitor acts like a break in the wire or “open circuit,” and the current is essentially zero. Now we “shut off” the battery (replace it with a wire). The capacitor will then release its charge, driving current through the circuit. In this case, the voltage across the capacitor and across the resistor are equal, and hence charge, voltage and current all do the same thing, decreasing with time. As you saw in class, this decay is exponential, characterized by a time constant t, as pictured in fig. 3.V0/e = 0.368 V0t = τVR,0=VC,0=ε; I0=ε/R; Q0 = CεVR, VC, I, QTimeFigure 3 Once (a) the battery is “turned off,” the voltages across the capacitor and resistor, and hence the charge on the capacitor and current in the circuit all (b) decay exponentially. The time constant is how long it takes for a value to drop by e (~2.7).E08-2(a)(b)(a)(b)The Details: Measuring the Time Constant In this lab you will be faced with an exponentially decaying current I = I0 exp(-t/) and an exponentially decaying voltage V = V0 exp(-t/). From data of voltage vs. time (or current vs. time)s, you will want to extract the time constant . We will do this in two different ways, using the “two-point method” or the “logarithmic method,” depicted in Fig. 7. (t2, I2)(t1, I1)TimeTimeFigure 7 The (a) two-point and (b) logarithmic methods for measuring time constantsIn the two-point method (Fig. 7a) we choose two points on the curve (t1,I1) and (t2, I2). Because the current obeys an exponential decay, I = I0 exp(-t/), we can extract the time constant most easily by picking I2 such that I2 = I1/e. We should, in theory, be able to find this for any t1, as long as we don’t switch the battery off (or on) before enough time has passed. In practice the current will eventually get low enough that we won’t be able to accurately measure it. Having made this selection, = t2 – t1.In the logarithmic method (Fig. 7b) we fit a line to the natural log of the current plottedvs time and obtain the slope m, which will give us the time constant as follows:( )( )( )( )( )( )( )( )( )22 112 122 1 2 1 12 102 1 0 2 1 2 1ln lnrise 1lnrun1 1 1 1ln lntt ttI t I tI tmt t t t I tt tI eet t I e t t t tττττ τ−− −−−⎛
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