PHYS 102 Lab Report 4

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University of Illinois, Chicago *

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102

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Electrical Engineering

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Apr 3, 2024

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PHYS 102 Lab Activity 4 (RC Circuits) 2/27/2024 Noah Pogonitz Daniel Afshari Ramiro Alvarado Introduction: In our previous lab experiment exploring Ohm’s Law, we observed the relationship between resistance, voltage, and currents through building a circuit. In this lab experiment, we will add in a new part to our study of circuits: the capacitor. Capacitors are devices that store charge, and its capacitance is its ability to store charge. During this experiment, we will be determining the capacitance through the measurement of a capacitor from the capacitor kit in the lab room. After building our circuit and using the IOLab data, we can determine the voltage at a specific time point, and if we set tau equal to time we can determine the voltage. We know that tau = RC, so if we know tau and resistance of our resistor, we will be able to calculate the capacitance. We hypothesize that the more resistance that the resistor provides in the circuit, the longer time it will take to charge up the capacitor and the larger tau will be. Furthermore, we hypothesize that the value of tau will be directly proportional to both the capacitance and resistance. Also, at greater voltages, it will take less time to charge up the capacitor because there will be more energy per unit of charge with greater voltages, meaning the capacitor will take less time to charge. Another aspect we kept in mind was the capacitance of the capacitor. We observed from the capacitor lab simulation that as the area of the capacitor increases, so does the capacitance, and as the distance between the two plates decreases, the capacitance increases. Materials and Methods: To get started with the experiment, we obtained a breadboard, a capacitor, and a resistor. Utilizing these parts, we set up a circuit and connected our IOLab device to the breadboard. Using IOLab data obtained, we found the voltage and the subsequent time for the capacitor to charge up (tau). Using this time and the fact that tau = RC, we were now able to calculate the capacitance of the capacitor because C = tau/R. We built two circuits differing with the amount of resistance. In the first circuit, we used a resistance of 4,208 Ohms and in the second circuit we

used a resistance of 15,000 Ohms. Analysis & Results:

In both images: V0 = Peak of Analog 8 before drop

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Resistance: 4,208 Ω (with a 2% uncertainty) Table 1: circuit run with a resistor of 4,208 Ohms Trial # V0 (in volts) Time at V0 V (in volts) Time at V Tau 1 1.795 7.42296 0.6603805 8.322 0.89904 2 1.791 8.08819 0.6589089 8.99687 0.90868 3 1.787 7.78430 0.6574373 8.70362 0.92022 4 1.786 8.22 0.6570694 9.14 0.92 5 1.783 9.43 0.6412497 10.37 0.94 6 1.783 7.65 0.6559657 8.57 0.92 7 1.785 8.51 0.6567015 9.43 0.92 8 1.787 8.17 0.6574373 9.09 0.92 9 1.778 6.31 0.6541262 7.23 0.92 10 1.783 7.46 0.6559657 8.38 0.92 Calculated Capacitance Using Average Tau = τ/R = 0.918794 / 4208 = 218 +/- 7.958 μF Std. Dev = (0.92022-0.89904)/2 = 0.01059 Standard Error: 0.01059/ 3.16227766 = 0.03348852 Std. error of C = Tau/R =0.03348852 /4,208 = 7.958E-6 F Resistance: 15,000 Ω (with a 2% uncertainty) Table 2: circuit run at a resistor of 15,000 Ohms Trial # V0 (in volts) Time at V0 V (in volts) Time at V Tau 1 1.759 19.48364 0.6471361 22.60262 3.12 2 1.772 18.53 0.6519188 21.74 3.21 3 1.753 14.63 0.6449287 17.84 3.21

4 1.768 16.9937 0.6504472 20.2151 3.2214 5 1.769 17.08383 0.6508151 20.29267 3.20884 6 1.754 15.43317 0.6452966 18.64175 3.20858 7 1.78 20.83 0.654862 24.04 3.21 8 1.769 16.77391 0.6508151 19.97275 3.19884 9 1.764 16.89 0.6489756 20.09 3.2 10 1.771 17.58431 0.6515509 20.79328 3.20897 Calculated Capacitance Using Average Tau = τ/R = 3.199663 / 15000 = 213+/- 1.069 μF Std. Dev = (3.2214-3.12)/2 = 0.0507 Standard Error: 0.0507/ 3.16227766 = 0.016032748 Std. error of C = Tau/R = 0.016032748/15,000 = 1.069E-6 F Average Calculated Capacitance = 215.8 μF Measured Capacitance by Manufacturer = 220 μF Discussion: When analyzing the simulation we analyzed that the battery would charge the capacitor, who’s maximum charge was dependent on both the distance, and area of its plates, which would then go on to power the lightbulb when the circuit was changed to the RC circuit. From this we used the knowledge that capacitors will charge up until full, upon which they will stop current flow, and supply the circuit with voltage once the battery output is gone. This allowed us to build a circuit which would supply a capacitor with charge using the IOLab DAC output as a battery, and then turning it off once the capacitor was fully charged, which, when turned off, the capacitor was allowed to supply the circuit. Using these premises we conducted an experiment that revolved around finding the relationship between the charge build up and charge decay of a capacitor which would both have the same constant, τ = RC, which is the estimate, but to ensure we get an accurate result, we will be using the equation V=V0e^(-t/τ). Here V equals to Voltage

after time which is (t), and V0 is the initial voltage of capacitor, which we measured at the maximum, all of which was attained using the IOLab and used to find unknown τ. This would then further allow us to find the capacitance of the capacitor using the equation C = τ/R derived from the first equation τ = RC. The uncertainty of this experiment is due to the IOLab only being able to read a certain number of digits, so there were many times we had to round to the nearest V and thus time. This would lead to a systematic error because all measurements are being affected in the same way, which would be being off by a couple of significant figures. As a group, we had trouble finding, and using resistors that allowed our data to flow the natural format of the experiment. For example, we needed to start our first experiment with a τ estimate of about 1 second which would come from the equation τ = RC. We had to chain many resistors to get a resistance of about 4,206 Ohms, and a capacitor with 220μf for capacitor value, which gave us around 1 second. Aside from this it was somewhat tricky for us to figure out the set up and the values to calculate for this experiment, but it was a rather trivial problem compared to the set up. This experiment helped strengthen our understanding of RC circuits as we found the relationship between Voltage, Capacitance, and τ which also helped us find the rate of charge and decay for capacitors. Conclusion: The experiment supported our hypothesis that the capacitance and resistance are both directly proportional to the time constant tau, and in turn that the resistance and capacitance are inversely related, as when we increased the resistance, the value for tau increased by a proportional amount. Furthermore, since the capacitor was constant throughout the trials, we were able to find an accurate value for the capacitance by using the equation τ=RC, further supporting the direct relation of the variables. The data we presented was accurate with minimal error, as the two calculated capacitance values using different resistances had overlapping ranges (218 +/- 7.958 and 213+/- 1.069 μF). Additionally, our calculated average capacitance (215 μF) was lower than the manufacturer’s capacitance (220μF), but this is explained by the fact that the capacitance may decrease overtime with use, meaning the value we obtained is likely explained by degradation overtime. In the future, we can use resistors with more accuracy and a device with more measurements over time than the IOLab to obtain more accurate values. Furthermore, we can conduct a future experiment to see how the capacitance of a capacitor is affected by degradation in which we test a capacitor at different amounts of usage to see if the capacitance degrades. Contributions: Daniel - Results & Conclusion Ramiro - Discussion & Analysis Noah - Introduction & Materials/Methods

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