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Experiment 6 The RC Series Circuit The voltage across the resistor and capacitor is an RC series circuit are examined using an oscilloscope. By measuring the voltage across either the resistor or capacitor at various times, the time constant of the circuit is determined. Theory, Charging A resistor and an uncharged capacitor are connected in series to a battery of emf Vo and a switch, S. (Refer to Figure 1.) When the switch is closed at time t = 0, with no initial charge on the capacitor, current will begin to flow in a clockwise direction around the circuit. In this case the charge on the capacitor is gradually increasing. It can be shown that after a time t has elapsed, the charge on the capacitor is q CVo (1 e t / RC ). Figure 1. The RC series circuit. (1) The voltage across the capacitor, which is q/C, is then Vc = Vo(1 - e-t/RC). (2) Equation (2) indicates that the voltage across the capacitor starts at zero and builds up exponentially to a final value Vo. At a time t=RC, Vc reaches a value that is 63% of Vo. This term is referred to as the time constant of the RC circuit. That is, = RC. (3) After a time equal to three time constants the voltage across the capacitor reaches a value that is 95% of Vo, and after five time constants the voltage is 99.3% of Vo, a value that is essentially Vo. The current in the circuit starts at a maximum value but then decreases as the capacitor becomes charged. The decrease is, in fact, an exponential decrease which can be written as V I 0 e-t/RC. R 6-1 (4) The voltage across the resistor, which is the product of current and resistance, then becomes VR = Vo e -t/RC. (5) This expression shows that the voltage across the resistor decreases to 37% of it initial value, Vo after a time equal to one time constant, where the time constant is again =RC. After a long enough time elapses, the capacitor will be fully charged and the voltage across the capacitor will be Vo. Theory, Discharging Suppose now the capacitor has an initial charge Qo (=CVo), the switch is closed, and the battery is removed from the circuit. (Refer to Figure 2.) The voltage across the capacitor will decrease as the positive and negative charges leave the plates of the capacitor and neutralize each other. This is referred to as the discharging situation and the voltage across the capacitor is VC = Vo e-t/RC. Figure 2. The RC series circuit. (6) (Time. t = 0 is assumed to at the time when the capacitor first begins to discharge.) Because current is flowing in the opposite direction from when it was charging, the voltage across the resistor starts out negative and gradually reaches zero as the current decreases. The voltage across the resistor in the discharging situation is then VR = Vo e-t/RC. (7) After a time equal to one time constant, Vc decreases from Vo to 37% of Vo, and VR increases from -Vo to 37% of –Vo. Graphs of the voltage delivered by the battery and the voltages across the capacitor and resistor for the entire charging and discharging sequences of the RC series circuit are shown in Figure 3. In the experiment, an oscilloscope is used to measure and observe the voltages, and a signal generator set in the square wave mode acts as a battery delivering a voltage Vo, in the 6-2 charging cycle and a short in the discharging cycle. Because the signal generator has an output resistance, RG that is in series with the circuit, a more realistic description of the circuit is shown in Figure 4.For this situation, the voltages Vc and VR corresponding to the charging situation are VC = Vo [1- e-t/( RG +R )C], (8) Figure 3. The voltage delivered by the battery (V), the voltage across the capacitor (Vc), and the voltage across the resistor (VR) for the RC series circuit. VoR -t/( RG +R )C. VR e RG R' (9) In the discharging situation, the voltages are VC = Vo e-t/( RG +R )C, (10) and VoR -t/( RG +R )C. VR e RG R' 6-3 (11) Notice that the time constant is now = (RG+R) C. (12) Figure 4. The circuit components connected to measure the voltage across (a) the capacitor and (b) the resistor. Apparatus o oscilloscope with leads o signal generator o 3 leads o decade resistance box, set at 1000 , ±5 o capacitance box, set at 0. 1 5 F, ± 0. 0 1 5 F Procedure 1) Connect the circuit elements as shown in Figure 4(a). Set the voltage output knob on the signal generator to I volt. At this setting the output resistance of the signal generator, RG, is 52 ohms. 2) Turn on the oscilloscope and the signal generator. Adjust the horizontal sweep knob (TIME/DIV) and the frequency of the signal generator until a series of charging curves and discharging curves for the capacitor are seen on the screen. Be sure that the starting point and the flat portion of the curves are seen. (Refer to Figure 5.) 3) Adjust the amplitude knob on the signal generator until the vertical distance between the beginning of the charging curve and the flat portion of the curve is 8 divisions high. 4) Adjust the horizontal sweep (TIME/DIV) until the charging curve is expanded so that the starting point and the 63% point fill horizontally as much of the screen as possible. (A factor of five expansions can be obtained by pulling out the horizontal position knob. When this is done, the horizontal sweep speed-reading is divided by five.) 6-4 Figure 5. The voltage across the capacitor when charging. 5) Determine the time constant of the circuit from the voltage curve for the capacitor that is observed on the oscilloscope. 6) Disconnect and reconnect the circuit according to the circuit diagram in Figure 4(b). The voltage across the resistor 'II look similar to the curve in Figure 6. Figure 6. The voltage across the resistor when the capacitor charging, then discharging. 7) Adjust the oscilloscope and the signal generator until the flat portion of the voltage curve is at zero volts. Expand the curve until the starting point and the 37% point fill horizontally as much of the screen as possible. 8) Measure the time constant, . 6-5 Analysis Three values of the time constant are found. The first one corresponds to the experimental value obtained from the voltage across the capacitor during the charging process, the second to the experimental value obtained from the voltage across the resistor during the charging of the resistor, and the third to the theoretical value obtained from (12). Report the three values of the time constant in a results table and calculate the percentage error of the two experimental values. 6-6