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Electrical Engineering

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Jan 9, 2024

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Table 5: Alexa Cumming ac75676 Luis Ainslie lma2862 Part 1. Methods: In this lab we will be testing the the scientist model of . V 1 is the voltage from the first 𝑉 1 𝑁 1 = 𝑉 2 𝑁 2 circuit and V 2 is the voltage induced in the second circuit. N 1 corresponds to the number of loops of the wire in the first circuit and N 2 is the number of loops of the wire in the second circuit. Based on this experiment, we are keeping the N values (loops in wire) constant as we are keeping the inductors constant. This equation can be expressed linearly as follows: V 2 = ( V 1 . The the value is known as 𝑁 1 𝑁 2 ) 𝑁 1 𝑁 2 the coil ratio for our given transformer. Based on this setup, we are predicting that there will be a small voltage in the second circuit due to the induction from the primary coil and there being more coils in the larger inductor. We will be testing this by gathering the voltage values in the digital multimeter and changing the voltage offset in the PASCO voltage. We will verify if our hypothesis is correct by using a t-score between the expected voltage and the experimentally measured voltage. Additionally, we will be testing if the “linear” model presented above is accurate in describing the relationship between the variables by using a chi-squared goodness of fit test. To conclude part one, and finish our analysis of the scientists model we will determine a coil ratio from our experimental data using a weighted average slope and weighted average slope uncertainty. We will then compare our experimental results with the given theoretical value of the ratio of 0.003117 ± 0.000002. In order to minimize our uncertainty we will be conducting multiple trials over a wide range of amplitudes. Our uncertainty for the digital multimeter is 0.1 mV and the PASCO software is 500mV due to being a digital device. Similar to our methods, Madelyn and Fran will also be using milliVolts and have the same goal of taking 10 trials to determine if there will be any voltage in the second circuit. Amplitude (mV) Initial Voltage Reading (mV) Experimental Voltage (mV) Hypothesized Voltage (mV) T-Score Coil Ratio 5000 16.1 22.8 22.5 0.0006 0.0045 6000 19.3 27.3 27.0 0.0006 0.0045 7000 22.5 31.8 31.5 0.0006 0.0045 8000 25.7 36.3 36.0 0.0006 0.0045 9000 28.9 40.9 40.5 0.0008 0.0045 10000 32.1 45.4 45.0 0.0008 0.0045 11000 35.3 49.9 49.5 0.0008 0.0045 12000 38.5 54.4 54.0 0.0008 0.0045 13000 41.8 59.1 58.5 0.0008 0.0045 14000 44.9 63.5 63.0 0.0010 0.0045 Table 1: Table showing the relationship between the V 1 and V 2 . The value used for the coil ratio is 0.0045 based on the chi-squared.

Table 5: Alexa Cumming ac75676 Luis Ainslie lma2862 Voltage (V) 22.8 27.3 31.8 36.3 40.9 45.4 49.9 54.4 59.1 63.5 Voltage based on Equation (V) 22.5 27.0 31.5 36 40.5 45 49.5 54 58.5 63 Chi-Squared Value 0.962 - indistinguishable Table 2: Table showing the current experimentally determined compared to the current determined by equation V 2 = 0.0045(V 1 ). Graph 1: Graph showing the relationship between voltage and amplitude. Sample Calculation Experimental Voltage: (this value represent V 2 ) 16. 1 ?𝑉 ( 2 ) = 22. 8 ?𝑉 Hypothesized Voltage: V 2 = ( V 1 = V 2 = 0.0045 (V 1 ) 𝑁 1 𝑁 2 ) V 2 = 0.0045 (5000mV) = 22.5 mV Chi-Squared Test: ? 2 = 1 𝑁 𝑖 = 1 𝑁 ∑ (? 𝑖 − ?(? 𝑖 )) 2 δ? 𝑖 2 + + + … = 0.925 ? 2 = 1 10 𝑖 = 1 10 ∑ (22.8−( 22.5)) 2 22.5 (27.3−( 27.0)) 2 27.0 (31.8−( 31.5)) 2 31.5 (36.3−(36.0)) 2 36.0 (63.5−( 63.0)) 2 63.0 x = 0.962 T Score of Voltage: = (indistinguishable) ? = 𝑃 1 − 𝑃 ??𝑖?ℎ??? | | δ𝑃 1 2 + δ𝑃 ??𝑖?ℎ??? 2 22.5 − 22.8 | | 0.1 2 +500 2 = 0. 0006 < 1

Table 5: Alexa Cumming ac75676 Luis Ainslie lma2862 Weighted Average and Uncertainty of Coil Ratio: Coil Ratio: = = 𝑉 2 𝑉 1 𝑁 1 𝑁 2 22.5 22.8 = 0.986 𝑁 1 𝑁 2 𝑃 ??𝑖?ℎ??? = 𝑃 1 δ𝑃 1 + 𝑃 2 δ𝑃 2 + 𝑃 3 δ𝑃 3 + 𝑃 4 δ𝑃 4 +𝑃 5 δ𝑃 5 δ𝑃 1 +δ𝑃 2 +δ𝑃 3 +δ𝑃 4 +δ𝑃 5 = (0.0045)(500) + (0.0045)(500)+(0.0045)(500)... +(0.0045)(500)+(0.0045)(500) (500)+(500)+(500)...+(500)+(500) 0.0045 = δ𝑃 = 500?𝑉 (??𝑖?? ℎ𝑖?ℎ??? ???????𝑖???) T Score of Coil Ratio: = ? = 𝑃 1 − 𝑃 ??𝑖?ℎ??? | | δ𝑃 1 2 + δ𝑃 ??𝑖?ℎ??? 2 0.0045− 0.003117 | | 500 2 +0.000002 2 = 0. 000003 < 1 𝑖??𝑖??𝑖???𝑖?ℎ???? Conclusion: Based on our results we were able to conclude that each of the three goals of the experiment were supported. First, we were able to verify that there was a voltage in our second circuit by comparing the value of the voltmeter to the expected value from the equation. In doing the t-test between the hypothesized and experimental voltage we concluded that the values were indistinguishable (t=0.0006 <1) and we can conclude that there was an induction from the first circuit to the second circuit. Next, we were able to conclude that our relationship was following a “linear” model and was accurate in describing the relationship between our equation V 2 = ( V 1. We were able to conclude this by doing a chi-squared test 𝑁 1 𝑁 2 ) of linear independence. Our calculated chi value was 0.962 (x<1) meaning that our values are indistinguishable and a linear relationship can be used to explain the relationship between both variables (V 1 , V 2 ). Lastly, we can conclude that the experimental coil ratio and the expected coil ration are indistinguishable. We were able to determine this by using a weighted average and finding a value to represent our coil ratio ( ). We then used the weighted average slope uncertainty to calculate a t-score 𝑁 1 𝑁 2 by comparing our value to the given value of the theoretical coil ration. Our t-score showed that t=0.000003 (t<1) and we were able to conclude that our results are indistinguishable. Based on our three goals being supported, we can say that we did a good job limiting the uncertainty of our experiment and conducted a sufficient number of trials. Overall, our results strongly support the validity of our model and our hypothesis. For the next iterations of our model, we could change the inductors to see if our results hold true (swap the connections and test to see if the linear model still holds true). Additionally, we could alter the efficiency of the transformer by adding a rod to investigate the effects on the coil ratio? Lastly, we could also test different voltage ranges to see if the results hold true for all values. Similarly, to Fran and Madelyn we found that the scientific model was supported in both cases. Part 2. In this part of the lab, we maintained both circuits but changed the wiring to measure the larger inductor's voltage using a multimeter with a systematic uncertainty of +/- 0.001V. We also inserted an iron rod to increase magnetism on coil 2 from coil 1 in a second iteration. It was anticipated that these modifications would not affect the linear relationship between the voltages and the consistency of the coil ratio, as predicted by the voltage model previously mentioned. To measure the voltage, we kept V 2 constant while

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Table 5: Alexa Cumming ac75676 Luis Ainslie lma2862 varying V 1 within a range of controlled values supplied by the PASCO machine controlled by PASCO software. We initially tried to adjust the maximum voltage setting to obtain more precise readings and reduce uncertainty but encountered issues, as the multimeter could not produce stable readings for V 2 . Consequently, we reverted to the recommended software settings from Part 1 but switched the measurement to volts. In an attempt to reduce uncertainties, we increased the number of experimental trials to ten. For Part 2, we used an uncertainty of 0.5 volts for the multimeter instead of the systematic uncertainty of the machine due to circuit noise, as recommended in the lab manual. To compare the theoretical and experimental results for voltage and coil ratio, we will use a t-test, similar to the statistical analysis method used in Part 1. Additionally, we will also use a chi-squared linear test to determine the linearity of the model with these changes. Wiring Change Amplitude (V) Initial Voltage Reading (V) Experimental Voltage (V) Hypothesized Voltage (V) T-Score Coil Ratio 1 0.512 0.724 0.876 0.298 1.2103 1.5 0.945 1.336 1.617 0.551 1.2103 2 1.310 1.853 2.242 0.763 1.2103 2.5 1.648 2.331 2.821 0.961 1.2103 3 2.052 2.902 3.512 1.196 1.2103 4 2.838 4.014 4.858 1.655 1.2103 5 3.890 5.501 6.658 2.269 1.2103 5.5 4.270 6.039 7.309 2.491 1.2103 6 4.680 6.619 8.01 2.726 1.2103 6.5 5.280 7.467 9.037 2.999 1.2103 Table 3: Table showing the relationship between the V 1 and V 2 . The value used for the coil ratio is 1.2103. Voltage (V) 0.724 1.336 1.853 2.331 2.902 4.014 5.501 6.039 6.619 7.467 Voltage based on Equation (V) 0.876 1.617 2.242 2.821 3.512 4.858 6.658 7.309 8.01 9.037 Chi-Squared Value 11.159 - distinguishable Table 4: Table showing the current experimentally determined compared to the current determined by equation V 2 = 1.2103(V 1 ).

Table 5: Alexa Cumming ac75676 Luis Ainslie lma2862 Graph 2: Linear relationship of Amplitude and Experimental Voltage collected with Inductor Reversal Experiment. Trial With Rod Inserted Amplitude (V) Initial Voltage Reading (V) Experimental Voltage (V) Hypothesized Voltage (V) T-Score Coil Ratio 1 2.461 3.480 12.793 18.264 3.6758 1.5 3.746 5.297 19.473 27.801 3.6758 1.8 4.527 6.402 23.533 33.597 3.6758 2.3 5.839 8.258 30.353 41.898 3.6758 2.5 6.356 8.989 33.040 47.168 3.6758 2.8 7.130 10.083 37.064 52.914 3.6758 3 7.650 10.819 39.768 56.774 3.6758 3.3 8.440 11.936 43.874 62.636 3.6758 3.5 8.950 12.657 46.525 66.421 3.6758 4 10.251 14.496 53.283 76.067 3.6758 Table 5: Table showing the relationship between the V 1 and V 2 . The value used for the coil ratio is 3.6758. Voltage (V) 3.48 5.297 6.402 8.258 8.989 10.083 10.819 11.936 12.657 14.496 Voltage based on Equation (V) 12.793 19.473 23.533 30.353 33.04 37.064 39.768 43.874 46.525 53.283

Table 5: Alexa Cumming ac75676 Luis Ainslie lma2862 Chi-Squared Value 87.645 - distinguishable Table 6: Table showing the current experimentally determined compared to the current determined by equation V 2 = 3.6758(V 1 ). Graph 3: Linear relationship of Amplitude and Experimental Voltage in the Inductor with Metal Rod Experiment. Sample Calculation Experimental Voltage: (this value represent V 2 ) ?𝑉 ( 2 ) = ?𝑉 Hypothesized Voltage: V 2 = ( V 1 = V 2 = 1.2103 (V 1 ) 𝑁 1 𝑁 2 ) Chi-Squared Test: ? 2 = 1 𝑁 𝑖 = 1 𝑁 ∑ (? 𝑖 − ?(? 𝑖 )) 2 δ? 𝑖 2 + + + … = 11.159 ? 2 = 1 10 𝑖 = 1 10 ∑ (0.876−(0.724 )) 2 0.724 (1.336−(1.617)) 2 1.617 (1.853−(2.242 )) 2 2.242 (2.331−(2.821)) 2 2.821 (2.902−(3.512 )) 2 3.512 T Score of Voltage: = (indistinguishable) ? = 𝑃 1 − 𝑃 ??𝑖?ℎ??? | | δ𝑃 1 2 + δ𝑃 ??𝑖?ℎ??? 2 0.876 −0.724 | | 0.1 2 +0.5 2 = 0. 298 < 1 Weighted Average and Uncertainty of Coil Ratio: Coil Ratio: = = 𝑉 2 𝑉 1 𝑁 1 𝑁 2

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Table 5: Alexa Cumming ac75676 Luis Ainslie lma2862 = 1.2103 𝑁 1 𝑁 2 = 1.2103 𝑃 ??𝑖?ℎ??? = 𝑃 1 δ𝑃 1 + 𝑃 2 δ𝑃 2 + 𝑃 3 δ𝑃 3 + 𝑃 4 δ𝑃 4 +𝑃 5 δ𝑃 5 δ𝑃 1 +δ𝑃 2 +δ𝑃 3 +δ𝑃 4 +δ𝑃 5 = (1.2103)(.5) + (1.2103)(.5)+(1.2103)(.5)... +(1.2103)(.5)+(1.2103)(.5) (.5)+(.5)+(.5)...+(.5)+(.5) 0.00 = T Score of Coil Ratio: With wiring switched: = ? = 𝑃 1 − 𝑃 ??𝑖?ℎ??? | | δ𝑃 1 2 + δ𝑃 ??𝑖?ℎ??? 2 0.003117− 1.2103 | | 0.5+0.00002 = 2. 414 > 1 𝑖????????𝑖?? With rod inside: 0.003117− 3.678 | | 0.5 2 +0.00002 2 = 6 = 7. 35 > 1 ?𝑖??𝑖???𝑖?ℎ???? Conclusion: In this second part of the experiment, we proposed the concept that inversing the two coils used in our experiments would not change the linear relationship of the amplitude and experimental voltage gathered, as shown in Part 1. After calculating the necessary Chi-Square Tests and T-scores for both the amplitudes, voltages, and coils, we gathered that there were distinguishable changes regarding said measurements. For the Chi-Square test regarding the coils themselves, we calculated a value of 11.159, which is greater than 1, thus proving that there isn’t a linear relationship now with the smaller coil being the inductor compared to the larger instrument. As for the Chi-Square test with the metal rod incorporated with the coils, we gathered a Chi-Square Test of 87.645, which once again, was distinguishable according to the test’s rulings. When determining the T-scores of the coils, we noticed that anything less than 3 Volts for the amplitude would give an indistinguishable value less than 1, but the majority of measurements with amplitudes past 3 Volts were inconclusive, and bridged even distinguishable results when considering out last value of 6.5 Volts. With the T-scores of the coils with the metal rod inserted, all of the measurements acquired were past the value of 3, thus resulting in distinguishable values for this experiment. Ultimately, our data allowed us to conclude that such changes regarding a smaller coil as an inductor, compared to a larger coil, and the use of a metal rod as an amplifier to the voltage would drastically change the linear relationship regarding amplitude and voltage in such a circuit. Possible sources of error in the experiment included an incorrect setup of the circuit and resulting in inaccurate readings on the Multimeter. Additionally, the model may have been flawed, failing to account for a coil ratio beyond a particular threshold due to increased energy loss caused by resistance. For future replication of these experiments, one may use equally-sized coils with regard to their resistance, as this could lessen the T-score values gathered, and present a linear relationship with respect to amplitude and voltage. Another revision to this part of the experiment could be the regulation of wiring when assembling the circuit incorporated, which would ensure that the cation and anion particles are flowing to the positive and negative sides of each coil/instrument.

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