Experiment 17 - Declan Rogers

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Jan 9, 2024

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Report for Experiment #17 DC Circuits Declan Rogers Lab Partner: Aiden Kaneshiro TA: Zhuyao Wang 6-4-22 Abstract This experiment is designed to examine the interaction of circuits with respect to theoretical expectations determined using Ohm’s law and general understanding of circuits. Actual values were found for each component in the circuits including two batteries and two resistors and they were then compared to the calculated values of these components determined through experimentation and data analysis. First the batteries were examined, and their voltages were calculated with the corresponding error. Then resistors were placed in the circuit to fully examine the behavior of Ohm’s Law in the practical world and how it changes by scenario. Finally, multiple resistors were utilized to compare power across components alongside the differences in series and parallel circuits with resistors. Each scenario was extremely precise and accurate to the expected theoretical value, falling quite close to the actual and nominal values of

the respective components. All instances performed well, and the experiment was a huge success in exemplifying the interaction of voltage, current, resistance, and power in dc circuits. Introduction This experiment analyzed the behavior of direct current and the attributes it displays in different circuits. The current, resistance, voltage, and power of the circuits were the values specifically observed. The first investigation utilizes two batteries to examine how voltage changes depending on if the batteries are in series or parallel with one another. Voltage values were then gathered in both instances. Voltage was also gathered in isolation to determine the actual voltage of each battery. The second investigation gathered resistance values through the current and voltage. The battery was attached to a 100 Ω resistor, to gather the data on voltage and current. To set this up, the resistor was disconnected from the battery and then reconnected with the voltmeter and ammeter placed in the circuit to determine the current and voltage. Those values were then used to determine the calculated value of the resistance to compare to the actual value found before. The burden voltage of the battery was then determined to solve for the internal resistance found in the battery through the circuit’s current value. The third investigation focused on the measurement of current and voltage across resistors in both series and parallel orientations. As in the second investigation, the actual values of the resistors were first recorded, and then they were placed into the circuits. In the instance with resistors in series, the current of the whole circuit was determined, and then the voltage across each component as current is constant in series. The voltage values were then used to determine the total voltage as well as the voltage drop. The current and resistance values were then used to compute power across the resistors in the circuit, using the definition that power is work done per unit time on that specific component. Finally, the overall resistance was determined through the total voltage and current to contrast it to the measured resistance. In the second instance of the investigation, only one voltage value was gathered while the currents for each resistor were taken because, in parallel, the current is different between the components, but the voltage stays the same across the different branches. Those differing current values were used to determine the power over each resistor. All three investigations exemplify how the equations used in circuitry can be applied to practical electrical work. The experiment was extremely successful in demonstrating the real examples of how circuit components interact with regard to these equations. Investigation 1 The first investigation begins by starting the DMM and setting it to the voltmeter. Connect one of the wires to the negatively charged lead and another wire to the positively charged lead that is labeled for resistance and voltage. Attach alligator clips to these wires and to both ends of two more wires.

Once setup is complete, begin by attaching the two wires of the voltmeter to the corresponding leads of the battery (positive to positive, negative to negative). Record this displayed value of the battery, swap the wires, and record that value as well. Use one of the free wires with alligator clips and connect one end to the positive of one of the batteries and the other to the negative end of the other battery. Then take the negative lead of the voltmeter and attach it to the free negative end of one of the batteries and connect the positive lead of the voltmeter to the free positive end of one of the batteries. Record the value displayed for the batteries in series. Then set the batteries in parallel by connecting both of their positive ends together and both of their negative ends together, respectively, using alligator clips. Then attach the voltmeter leads to the positive and negative ends of one of the batteries. Record the voltage of the batteries in parallel displayed on the voltmeter. Fig. 1: Circuit diagram for one battery and voltmeter in parallel Fig. 2: Circuit diagram for two batteries in series and voltmeter in parallel

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Fig. 3: Circuit diagram from two batteries and voltmeter in parallel Table 1: Voltage for one battery, two batteries in series, and two batteries in parallel V d V Battery DMM- 1.5176 0.01517 6 Battery DMM+ - 1.5176 - 0.01517 6 Battery DMM- 1.5164 0.01516 4 Battery DMM+ - 1.5164 0.01516 4 Series 3.044 0.03044 Parallel 1.5168 0.01516 8 This table displays the values found from the investigation following the steps outlined above. To determine the error in the voltages found, the equation below was utilized as 1% error in voltage was given. δV = V ∗ 0.01 The results found in this investigation reflect the expected performance of the circuits as the batteries used (D batteries) generally carry a voltage of 1.5V. Furthermore, because the voltage stays constant in parallel it was expected that the read value would stay roughly the same. Likewise, because voltage adds when in series, it was expected that the voltage would roughly double, which the results support.

Investigation 2 For investigation two, do not change the setup of the voltmeter from investigation one. Take the second DMM and utilize it as an ammeter by setting the negative wire in the same spot as the voltmeter but place the positive wire in the positive port with 500mA shown. Use two more wires to connect to opposite ends of the 100-ohm resistor. Change the voltmeter to the ohmmeter settings because this is the first data set collected in this investigation. Start by collecting the actual resistance of the resistor with the ohmmeter and documenting it. Take on e of the batteries from the first investigation as the power source of this investigation. Connect it to the resistor and switch the ohmmeter back to the voltmeter mode. Ensure to connect it in parallel with the resistor. While the voltmeter is connected, connect the ammeter in series with the battery and resistor. Document the voltage displayed across the resistor and the current of the whole circuit. Fig. 4: Resistor, battery, and ammeter in series with voltmeter in parallel across the resistor

Fig. 5: Resistor, battery and ammeter in series Table 2: Measured ohm, voltage, and current values and errors of each Ω δ Ω V d V d V/V I d I d I/I 1 batter y 98.8700 0 0.9887 0 1.4825 0 0.0148 3 0.0100 0 0.0150 8 0.00015 0.0100 0 Because all errors were given as 1%, the absolute error is determined with the following equations: Equation 2: δ Ω = Ω ∗ 0.01 Equation 3: δ V = V ∗ 0.01 Equation 4: δ I = I ∗ 0.01 To find the relative error in voltage and current these equations were used: Equation 5: δ V V

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Equation 6: δI I Table 3: Calculated resistance, burden voltage, and internal resistance values and errors of each R d R d R/R burden V internal R d internal R 98.28295 1.38993 0.01414 0.01849 2.32697 0.60451 To determine the resistance to compare to the actual resistance from table 2, this equation will be used, derived from Ohm’s Law: Equation 7: R = V I The relative error of the resistance calculated above is determined with the equation below: Equation 8: δR R = √ ( δV V ) 2 + ( δI I ) 2 With this value, absolute error can be determined using the following equation: Equation 9: δR = ( R ) ( δR R ) Despite the fact that the burden voltage was measured, the internal resistance must be determined using the equation below and the voltage of the battery from investigation one with the measured voltage and current found in investigation two:

Equation 10: R Internal = DMM − V I The absolute error of the internal resistance is found using equation 11 below: Equation 11: δ R Internal = √ ( DMM 2 + δV 2 ) ( DMM − V ) 2 + δI I Following the calculation of the resistance using the measured voltage and current, it is shown that the results found in the experiment are consistent with the expected outcome of the experiment. The actual resistance of the 100-ohm resistor was found to be 98.87000 ohms and the calculated resistance was 98.28295 ohms, both in extreme proximity to one another and incredibly close to the nominal value. The internal resistance of the battery also lined up with the expectations of the experiment. The calculated internal resistance oft eh battery was 2.32697 ohms which is proportionally small compared to the circuit and is proximal to the expected resistance of near zero. Most of the errors in this experiment were quite small and negligible as the determined values were incredibly accurate to the predicted outcome. Investigation 3 Investigation three is set up by first recording the actual resistance of the 1k ohm resistor and the 470-ohm resistor and documenting those values for later comparison. Switch the DMM from the ohmmeter to the voltmeter and set the other DMM as an ammeter as explained in investigation two. Use the voltmeter to ensure that the power supply is in direct current and that it is at 5 volts. Then wire the two resistors in series with one another for the first section of the investigation. Begin by measuring the voltage across the two resistors and the ammeter and documenting it. Recall that the voltmeter must be in parallel to collect measurements. Because the components are measured in series, measure the current across the two resistors (which should be constant in series) and document this. Following the gathering of this data, reset the circuit such that the two resistors and power supply are all in parallel with one another. Document the voltage of this circuit as well as the voltage across each component in the circuit. The voltage should stay constant as they are in parallel. Then use the ammeter to record the current of the circuit and the current of each branch in the parallel circuit and record those values.

Fig. 6: Resistors and power supply in series with DMMs to measure current and voltage Fig. 7: Resistors and power supply in parallel with DMMs to measure current and voltage Table 4: Given and actual resistance values for the resistors R1 R2 Given 470.0 1000.0 Actual R 465.4 1020.2 This table displays the given and actual resistor values of both resistors which wil be used later for comparison with calculated values.

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Table 5: Current and voltage values for the circuit in series, the voltage across each component, and the voltage drop I acr oss R1 an d R2 V acr oss R1 V acr oss R2 d V2 d V2 /V V acr oss me ter d V me ter d V/ V me ter V tot al V dr op 0.0 03 37 1.5 63 00 3.4 18 00 0.0 34 18 0.0 10 00 0.0 04 05 0.0 00 04 0.0 10 00 4.9 81 00 0.0 15 00 The absolute error in the voltages are determined by the following equation as the error was given as 1%: Equation 12: δV = V ∗ 0.01 To find the total voltage of this circuit use the following equation. Equation 13: V Total = V R 1 + V R 2 The voltage drop is found using the equation below: Equation 14: V Drop = 5 V − V R 1 − V R 2 − V Meter Table 6: Calculated power across each resistor and calculated total resistance value with errors for each P 1 d P1 P 2 d P2 R=V/I d R/R d R 0.00527 0.00013 0.01155 0.00028 1480.2377 0.01414 20.93372

4 The power for each resistor is determined using the actual resistor value and the recorded current in the equation below: Equation 15: P = I 2 R To find the actual resistance of the entire circuit the following equation was used. Equation 16: R Total = V Total I Total Because the relative error in the voltage and current are always 0.01, the equation below will calculate the relative error for the whole resistance calculation: Equation 17: δR R = √ 0.01 2 + 0.01 2 This equation below can be used o determine the absolute error in the calculated resistance value: Equation 18: δR = ( R ) ( δR R ) Following the calculation of both of these values, the absolute error of the power found for both resistors can be determined through the equation below: Equation 19:

δP = ( P ) ( √ 0.02 2 + δR R 2 ) Table 7: Voltage and current values for each resistor when in parallel V across R1, R2, meter V1 V2 V meter I 1 I 2 I total I 1 theoretica l I 2 theoretica l 4.98900 4.9950 0 4.9950 0 4.9950 0 0.0107 3 0.0044 7 0.0156 0 0.01064 0.00500 Because a majority of these values are measured, only one equation is required to determine the theoretical current across the components which is shown below: Equation 20: I theoretical = 5 V R Table 8: Calculated power values across each resistor and errors for each: P1 d P1 P2 d P2 0.05362 0.00131 0.02034 0.00050 In this instance, because the resistors are in parallel, the resistance and current will be different for each power value and the equation below was used to determine each power vlues for each resistor: Equation 21: P = I 2 R The error in the power is found using the equation below using the same relative error in resistance from the previous series circuit:

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Equation 22: δP = ( P ) ( √ 0.02 2 + δR R 2 ) The first section of this investigation requires that a few details be validated to confirm that the results are consistent with what is expected. The total voltage across both resistors was found to be 4.98100V which is incredibly close to 5V, extremely close to an ideal circumstance. In the series circuit, the total resistance was found to be 1480.23800 ohms, which is extremely close to the total resistance of 1470.00000 ohms. In the second section where the resistors were in parallel, the values returned were extremely precise. The measured current of the circuit was 0.01073 A as compared to the theoretical current of 0.01064 A. Similarly the current of the second resistor was measured to have a current of 0.00447 A while the theoretical was 0.00500 A. Conclusion The intention of this experiment was to analyze the different behaviors of circuits in parallel and series in the practical world and how that compared to theoretical expectations. These investigations were also intended to analyze Ohm’s Law and power found across components in the circuits in the real world. In order to do this, the first investigation gathered a number of values that would be utilized in further instances throughout the experiment to examine how voltage behaves across the different circuits. The second investigation included a resistor to examine the interaction of Ohm’s Law completely. The final investigation used two circuits to follow the same examination processes as the first two investigations but with power included as well. The experiment performed as expected, with the resulting values staying consistently close to the theoretical values. In the first investigation, the voltage was determined to be 1.5176 V, quite close to the standard voltage of a D battery of 1.5V. In the second investigation, the actual resistance of the 100-ohm resistor was 98.87000 ohms and the calculated resistance was 98.28295 ohms, all close in proximity to one another. The resistance of the battery was also incredibly small as expected, only 2.31697 ohms. In the third and final investigation, while the two resistors were in series the total voltage was found to be 4.98100 V, close to the 5V that the power supply gave out. The total resistance was close to the expected value of 1470.00000 ohms, coming to a total of 1480.23800 ohms. In the second circuit of the investigation, the calculated current was close to the expected theoretical amount, reading 0.1073 A compared to the theoretical 0.01064 A on the first resistor, and 0.00447 A compared to the 0.00500 A for the second resistor.

Despite the fact that there were few major errors across data collection in the experiment, there were a few ways that discrepancies could be cleaned up. Utilizing previously unused batteries for the first and second investigations would guarantee quality outcomes and that the batteries had full potential and maximum voltage. Using hard connections for the circuits as opposed to alligator clips and the like would also minimize error across the connections to enhance the quality of results and reduce any extraneous resistance. Questions 1. Because the 6V battery and the 1.5V flashlight cell are wired in series the voltages will add and the terminal voltage is 7.5V. 2. The resistance is larger while in series by a factor of four because parallel resistors amount to the reciprocal of the sum of the reciprocal resistor values, reaching a final value of r/2 while the resistor values add in series, resulting in a total resistance of 2r, four times the parallel value. 3. A 1.5V battery has internal resistance and if the current is 200 mA the resistance of the battery is 7.5 Ohms. 4. The resistance of the teapot is 8.067 Ohms. 5. The resistance between points A and B in the first circuit is r and in the second circuit it is 2R.