RC Circuits Lab Report

Objective: To examine the charging and discharging process of capacitors in a RC circuit and determine their time constants (τ). Physical Principle: All RC and DC circuits contain a source of electromotive force (battery), resistors, and capacitors. However, in contrast to DC circuits without capacitors, the currents of RC circuits take some time to reach their constant values. This time is the time needed to charge the capacitors within RC circuits and is known as the time constant τ. For a basic RC circuit containing one resistor of resistance ‘R’, one capacitor of capacitance ‘C’ and one source of electro motive fore ‘Ɛ ’, the time constant can be expressed using the following equation: τ =R∙C During the charging process the voltage across the capacitor V c increases with time t from zero to Ɛ. The voltage across the capacitor can be determined using the following equation: [ V C =Ɛ 1−e ] (−tτ ) The dependence of the voltage across the capacitor on time during the charging process can be visualized by plotting a graph of the voltage across the capacitor versus time. When the charging time is equal to the time constant (t=τ), the voltage across the capacitor will be VC=0.63 Ɛ. When the source of the electro motive force is switched off, the voltage across the capacitor decrease as the capacitor discharges. Like the charging process, the time constant τ will be the product of the resistance and capacitance within the circuit. However, the charge across the capacitor must be determined using the following equation: [ ( )] V C =Ɛ e −t τ When the discharging time is equal to the time constant (t=τ), the voltage across the capacitor will be VC=0.37 Ɛ. Equipment: Two AA batteries with holder (battery Ɛ ) Digital multimeter Snap-circuit boards Two Snap-circuit resistors (10kΩ) Two Snap-circuit capacitors (470µF and 100µF) Snap-circuit SPDT switch Snap-circuit connectors Labquest2 interface and LoggerPro software Vernier voltage sensor Procedure: The apparatus was set up as described in the PHY Lab Manual, pages 48-49. A digital multimeter was first used to determine the exact values of resistance and capacitance to be used I the experiment. Using circuit one, the switch was set to the discharging position and the voltage probe zeroed. The trigger tab was then selected followed by selecting increasing in the triggering box and a trigger value of 0.005V was set. The collect button was then selected and the immediately changed to the charging position. The data was automatically recorded on a graph and stored by selecting ‘experiment’ followed by ‘store last run’. With the switch still in the charging position, the triggering tab was selected through ‘data collection’. ‘Decreasing’ was then selected and a trigger value 0.02V lower than the voltage stored on the capacitor entered. The collect button was then selected, and the switch immediately set to the discharging position. The data was automatically recorded on a graph and stored by selecting ‘experiment’ followed by ‘store last run’. The procedure was then repeated using circuits two and three after adding or removing the appropriate capacitor or resistor. All charging and discharging graphs were then divided and labelled. Lab Data: Please see the attached data sheets. Calculations: 10kΩ ,R 4 a=9.87kΩ10kΩ ,R 4 b =10. 20kΩ 100µF=118µ F470µF=501µF τ =CR Circuit 1 Charging Discharging −6 3 τWhen τ=t 5.05 5.11 τ Theoretical=CR=501×10 F∙9.87×10 Ω=4.95 5.11−5.05∨ ×100=1.2% 5.08 × 100= τ FIT−τ τ=t ∨ τ avg %Error= τFIT 5.11 5.16 5.08− 4.95∨ 4.95 × 100=2.6% τ avg−τ theoretical ∨ τ theoretical %Diffefence= ×100= Circuit 2 Charging Discharging τWhen τ=t 10.30 10.51 τFIT 10.37 10.52 τ =CR=501× 10−6 F∙(9.87× 103 Ω+ 10.2 × 10 3 Ω)=10. 06 ×100=0.7% 10.34 τ FIT−τ τ=t ∨ × 100= τ avg %Error= 10.37−10.30 ∨ 10.34 −10.06∨ 10.06 τ avg−τ theoretical ∨ × 100=2.8% τ theoretical %Diffefence= ×100= Circuit 3 Charging Discharging τ =CR=118×10−6 F ∙9.87 × 103 Ω=1.1 1. 53−1.52∨ τ FIT−τ τ=t ∨ 1.53 × 100=0.7% τ avg %Error= × 100= τWhen τ=t 1.43 1.52 τFIT 1.44 1.53 1.53−1.17 ∨ 1.17 τ avg−τ theoretical ∨ ×100=30.7% τ theoretical %Diffefence= ×100= Discussion: In this experiment, RC circuits consisting of a source of electromotive force, resistors, connecting wires and capacitors were used to determine the time required to completely charge or discharge specific capacitors. This time is directly proportional to the product of the capacitance and resistance present within the circuit and is known as the time constant τ. Theoretically this time constant can be calculated using the formula τ=CR. The time constant can also be calculated when the time constant is equal to the charging or discharging time. During charging, the time constant can be calculated by multiplying 0.63(1e-1) by the voltage from the battery( Ɛ=2.93V). During the discharging process, the time constant can be calculated by multiplying 0.37(e-1) by 2.93V. The time constant can also be extrapolated from a plot of voltage vs time. Circuit 1 was expected to produce a time constant of 4.95, however, from the graph a time constant of 5.11 was found along with a time constant of 5.05 when τ=t. The experimental values had a slight percentage error of 1.2%and a percentage difference from the theoretical value of 2.6%. Circuit 2 was expected to produce a time constant of 10.06 which is almost twice that of circuit 1. This was expected since circuit 2 has twice the resistance. From the graph a time constant of 10.37 was found along with a time constant of 10.30 when τ=t. The experimental values had a slight percentage error of 0.7%and a percentage difference from the theoretical value of 2.8%. Circuit 3 was expected to produce a time constant of 1.10, however, from the graph a time constant of 1.44 was found along with a time constant of 1.43 when τ=t. The experimental values had a slight percentage error of 0.7% and a percentage difference from the theoretical value of 30.7%. There was also a noted difference between the charging and discharging time constant values. These differences may have been caused by errors which occurred throughout the experiment such as systematic errors associated with the instruments used throughout the experiment. Also, the internal resistance of the ammeter may have also affected the time constant of the capacitor but was not accounted for. Gross errors may have also contributed to the errors which occurred throughout this experiment while readings were being recorded. Conclusion: The time constants of capacitors from three RC circuits were successfully determined experimentally and compared to their theoretical values which were determined mathematically. Answers to Questions: 1. Compare the experimental time constants measured in the charging and discharging processes. Which value is greater and why is it greater? The discharging experimental time constants were observed to be greater than the charging time constants 2. Does the time constant depend on the voltage delivered by the battery? The time constant does not depend on the voltage of the battery only the capacitance and resistance within the circuit. 3. Based on the parameters of the experiment determine the maximum charge accumulated on the capacitor. Using Circuit 1: [ V C =Ɛ 1−e ] (−tτ ) =2.93 [ 1−e −60 5.08 ]=2.93 [ 1−7.42 × 10−6 ] =2.93V 4. Based on the parameters of the experiment determine the maximum current flowing through the resistor. I= V = 2.93 =2.97×10−4 A R 9.87× 103 5. For this experiment, show a simple way of measuring the resistance of the multimeter. Connect the multimeter in series with and existing resistor, determine the time constant from the graph and use the following equation to determine resistance of the ammeter; τ =C ( R4 + R multimeter) R multimeter= τ −R 4 C