Seminar assignments - Rc circuits abstract and discussion PDF

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RC Circuits Abstract and Discussion...

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Abstract The purpose of this experiment was to study RC circuits. In part 1, we used three different setups: one capacitor, two capacitors in parallel, and two capacitors in series. Using these circuits, we calculated the time for the circuit to charge and discharge and used this data to find the time constant, the length of time that a transient current exists in an RC circuit, using the equation T=tht/ln(2). For the first setup, we compared this value to the theoretical T calculated from the equation T=RC. For the second and third setups, we used this equation to calculate the experimental values of capacitance and compared them to the theoretical values. Our percent differences varied greatly, from 10% to 160%, which will be discussed in the discussion. In part 2, we constructed a capacitor using aluminum foil and wax paper and used the RC circuit to determine the thickness of the paper. We found the thickness to be 6.9e-7, with 24% difference when compared to the value obtained by the micrometer. Generally, our data showed that as we increased the capacitance, the time constant also increased, which is consistent with the direct relationship illustrated by the equation T=RC.

Discussion In part 1 of this experiment, we evaluated the relationship between the resistance, the capacitance, and the rate of charge and discharge of the capacitors of 3 different RC circuits. As the capacitor charges, the flow of current through the capacitor decreases. When a circuit is closed, the rate of charge of a capacitor can be calculated using the equation T=RC. This equation illustrates a direct relationship between the time constant (T), or the length of time that a transient current exists in an RC circuit, and the capacitance, when R is held constant. The mathematical relationship between these variables can be found using the following equation: Vr(t)=-Voe-t/RC. Instead of using this equation during this experiment, we used an oscilloscope to read the graph, where the vertical scale was 1 V and the horizontal scale was 1e-4 s. We then drew the graph and calculate the half time (tht) for both charge and discharge. We were then able to calculate the time constant for charge and discharge from the equation, T=tht/ln(2) and average them to find our experimental time constant. After calculating the experimental value for T, we were able to compare these values to theoretical values, using the equation T=RC, when R was consistently 100 Ω. We performed this experiment for each of 3 different setups. For the first, our RC circuit was set up with only 1 capacitor and a resistor of 100 Ω. We then calculated the value for T and compared it to the theoretical value found from T=RC. Our theoretical value for T was 1e-4 and our experimental value was 1.1e-4, with a percent difference of 10%. For the second, our RC circuit was set up with 2 capacitors in parallel and a resistor of 100 Ω. We then calculated the value for T and used this to calculate an experimental value of capacitance (C). Our theoretical value for C was 2e-6 F and our experimental value was 1.3e-6 F, with a percent difference of 35%. This same procedure was used for the third setup, with 2 capacitors in series and a resistor of 100 Ω. Our theoretical value for C was 5.0e-7 F and our experimental value was 1.3e-6, with a

percent difference of 160%. These large percent differences between the theoretical and the experimental values could be due to ambiguities when reading the oscilloscope graph about where the line intercepted, leading to difference in the measured and actual half times and time constants. Our results for the first and second setups were consistent with T=RC in that T increased as C increased. However, with the third setup, the total C did not change between putting the 2 capacitors in series and putting them in parallel. This could have been due to several things, including the fact that one of the spring on the circuit board was loose, which may have altered the flow of current and thus increased or decreased our oscilloscope reading respectively. However, this would have been a systematic error and would have caused similar uncertainties in each of the different setups. It could have also been due to ambiguities in where lines crossed in the graphs of the oscilloscope as mentioned before. In order to be able to view the charge and discharge graphs from which we gained our data, we set the function generator to square wave. This is so that the charge and discharge portions of the graph would be clearly visible on the oscilloscope graph. We had to set the frequency properly so that the capacitor had enough time to fully charge and fully discharge before the square wave switches on or off. When the switch is on, the capacitor is charging to Vo and when the switch is off, the capacitor is discharging to 0. If the frequency was set too high, then the graph would not have had enough time to level off and would not have enough time to reach Vo or 0, depending on if the switch was on or off. If the frequency was reduced, then the graphs would have had a longer plateau, meaning that it would have had more time to charge and discharge. For part 2 of this experiment, we used an RC circuit to measure the thickness of wax paper used as a dielectric in a capacitor created from 2 pieces of aluminum foil. In order to calculate this, we measured the length and width of the aluminum to calculate the area of the aluminum foil and used a micrometer to measure the actual width. We measured the time constant in the same way as in part 1 of the experiment, and then used the equation T=RC to find the capacitance. The resistance used for this part of the experiment was 100 kΩ. From this data, we were able to find the capacitance and use the equation C=kɛA/d to find d, the distance between the 2 plates of the capacitor, or the width of the wax paper. We used the dielectric value of wax paper, 3.7 and a permissivity of free space (ɛ) of 8.85e-12 C2/N*m2. From this, we found the experimental value for the thickness of the wax paper to be 6.9e-7 m, with 24% difference from the value obtained by the micrometer. If the dielectric value had been larger than 3.7, then it would have caused the calculated thickness of the wax paper and the capacitance to increase because of the direct relationship between the two values. Having a higher dielectric constant would allow the capacitor to hold more charge for a longer period of time, which would effectively increase the time constant. If the dielectric were less than 3.7, however, it would have the opposite effect. In this case, the calculated thickness of the wax paper and capacitance would be lower because of the direct relationship between the two values. Having a lower dielectric constant would decrease the amount of charge able to be held, and the charge would be held for a shorter period of, thus decreasing the time constant.

Overall, with consideration for sources of error as mentioned below, our data is consistent with the relationships illustrated in the equation T=RC, and that the time constant is dependent on both resistance and capacitance. Therefore, our data supports basic principles of physics and theoretical predictions from the given equations. There are many sources of error for this experiment. One possible source of error for part 1 includes ambiguities when reading the oscilloscope graph, including where exactly the lines crossed. This error could have led to error in the calculation of the half time and the time constant, and led to larger percent differences than actually existed. Another source of error, as mentioned before, is that one of the springs on the circuit board was loose, which could have altered the flow of current and the time for charging, thus altering the calculated time constant. One possible source of error for part 2 includes only including one trial for the measurement of the thickness of the wax paper. There could have been uncertainty related to the instrument being used, or uncertainty related to random error associated with the way in which the student measures the wax paper. Another source of error includes the fact that the 2 pieces of aluminum foil used were not exactly the same dimensions, which could have altered the amount of charge stored in the capacitor....