ME 360 Lab 6 Report

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Jan 9, 2024

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ME 360 – Lab 6 Report College of Engineering, University of Alabama Dynamic Response Report ME 360-004 Wednesday, November 29 th , 2023 Prepared by: Kayla Shows

Table of Contents A. Cover Page B. Table of Contents 1. Objectives ……………………………………………………………………… . ……………………….…….. 3 2. Results ………………………………………………………………………………………………………….. 3 3. Conclusions …………………………………………………………………………………………………….. 7 4. Appendix ………………………………………………………………………………………… .. …………… 8

3 Objectives The objectives of this experiment were to evaluate the time constant tau, 𝜏 , of a resistor-capacitor circuit and the bandwidth omega, ω bw , of the same circuit. Evaluating the time constant and bandwidth allows for the inversely proportional relationship between the two to be made clear. Results Based on the square-wave test data, the experimental time-constant, 𝜏 exp can be determined by estimating the value of the voltage when it starts at 5 seconds, then find 36.8% of this value and estimate the time at which this value occurs. Then to solve for 𝜏 exp , it is simply the difference between the start time and the time of the 36.8% value. For figure 1 below, the estimated voltage was assumed to be 4.8. Shown below are the steps to calculating 𝜏 exp for figure 1. 4.8 𝑉 ∗ 0.368 = 1.7664 𝑉 4.8 𝑉 − 1.7664 𝑉 = 3.03 𝑉 From these calculations, it is known the time corresponding to the voltage 3.03 on the curve is what needs to be determined. Figure 1. Finding 𝜏 exp from a graph After estimating that the second time value is 5.14, 𝜏 exp can be determined by subtracting this value from the start time as shown below τ exp = 5.14 − 5 = 0.14 𝑠?????𝑠 4.8 5.14 3.03

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4 The theoretical time-constant, 𝜏 th , can be calculated in order to compare the experimental and theoretical values of 𝜏 . 𝜏 th is calculated using the formula τ ?ℎ = 𝑅? (1) where R is the resistance in Ohms and C is the capacitance in Farads. Using equation (1) and that R=99.946 k Ω and C=1.02 μF from this experiment, τ ?ℎ is calculated as τ ?ℎ = 99,946 ∗ 1.02 ∗ 10 −6 = 0.102 𝑠?????𝑠 Next the gain, G, and phase, Φ, can be calculated for each of the harmonic input cases by using the following equations respectively 𝐺 = 𝑉 ??? /𝑉 𝑖? (2) Φ = tan −1 (−𝑋 ? /𝑅) (3) 𝑋 ? = 1/(2𝜋??) (4) where V out is the output voltage in Volts, V in is the input voltage in Volts, X c is the capacitive reactance in Ohms, R is the resistance in Ohms, f is the frequency in Hertz, and C is the capacitance in Farads. From equations (2) and (3) , gain and phase can be solved for each harmonic input case, the data for which has been placed in the appendix, entry 1. From this, a plot of linear gain vs linear frequency can be created for each harmonic input case as shown in figure 2 Figure 2. Plot of Linear gain and Linear Frequency While some of the gain values were over 1, which is not what was expected, the majority were below 1 and can be considered acceptable values. As with any experiment, some amount of error is expected, but these values above 1 skew the graph so that it does not look completely as expected. The last section, where all values for gain were below zero, show what type of graph was expected from this plot.

5 Since the linear frequency is known, a plot of linear phase against linear frequency can also be created Figure 3. Plot of Linear Phase and Linear Frequency This plot shows that as the frequency increases, the phase angle gets smaller, which is what was expected from this experiment. Now, there are four ways to estimate the bandwidth. The first is to plot the dB gain against logarithmic frequency and estimate the value based on where the high and low frequency asymptotes intersect, as shown in figure 4 below. dB gain can be calculated using the following formula 𝐺 ?𝐵 = 20𝑙?? 10 (𝑉 ??? /𝑉 𝑖? ) (4) again, the values for dB gain for each harmonic input cases can be found in the appendix, entry 1. Figure 4. Plot of dB Gain against Logarithmic Frequency

6 Looking at the point under where the two asymptotes intersect, without knowing exactly where that point is, the bandwidth is determined to be approximately 9 rad/sec. The second way is to find the -3dB point on the graph and estimate the value of the bandwidth from that, as shown in figure 4 above. From this analysis, the bandwidth is approximated to be 9.551 rad/sec, which is a slightly more accurate way to determine the bandwidth. Another way to solve for bandwidth is to plot phase against logarithmic frequency and estimate where the phase goes to -45 ° as shown in figure 6 below. Figure 6. Plot of phase and Logarithmic Frequency From this analysis it can be determined that the phase angle is roughly 45 ° when the frequency is 1.5 Hz. 9.551 Low Frequency High Frequency 1.5 -45 °

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7 Knowing that, the bandwidth can be determined by finding the linear gain value for that frequency in the appendix, table 1, and using equation (4) to solve for the dB gain, the answer for which should be close to -3dB. For example, shown below are the calculations for the 1.5 Hz frequency 𝐺 ?? = 20𝑙?? 10 (0.638) 𝐺 ?? = −3.90 ?? Finally, the last way to estimate the bandwidth is to use the relationship between ω bw and 𝜏 exp which can be shown as ω ?? = 1/τ ??? (5) Using this equation and that 𝜏 exp =0.102 sec from equation (1), ω ?? is found to be 9.80 rad/sec. Conclusions All the methods used in this experiment are roughly a good indicator for finding the bandwidth, ω ?? . Each method provided a reasonable value for the bandwidth and they were all close enough to each other that a range of between 9 rad/sec and 9.8 rad/sec can be determined for the bandwidth. The method that involves using equation (5) seems to be the best option in this case because, even though this equation uses the experimental value of 𝜏 , the experimental and calculated values of 𝜏 were very close to one another. Therefore, this would give a very reasonable and accurate value for the bandwidth. Another acceptable option is to find the -3dB point in figure 4. This method also gives a very reasonable and accurate calculation for the bandwidth. The only downside to this method is that sometimes the plot may be hard to read and find an exact value on, but nevertheless this method is still acceptable because it gives a reasonable value for the bandwidth. The other two options, involving the high/low frequency asymptotes and the phase vs logarithmic frequency plot, are less desirable because the plots may be hard to read and get exact values from. If the bandwidth estimation is to be as accurate as possible, the methods used need to be easy and accurate. Overall, the two preceding methods discussed are the best options in this case because they provide an easy way to solve for the bandwidth without sacrificing the accuracy needed when determining the values for this lab.

8 Appendix 1. Frequency Response Data Frequency (Hz) Input Amp. (V) Output Amp. (V) Gain (Linear) Gain (dB) Phase lag (deg) 0.02 2.984 2.988 1.001 0.00868 -89.266 0.1 4.827 4.928 1.020 0.172 -86.335 0.2 2.579 3.126 1.212 1.670 -82.699 0.4 2.660 1.442 0.542 -5.320 -75.629 0.5 1.383 0.026 0.067 -23.478 -72.241 0.6 2.073 0.330 0.159 -15.972 -68.977 0.7 4.776 0.049 0.010 -40.0 -65.849 0.72 4.624 3.048 0.659 -3.622 -65.241 0.75 4.436 4.531 1.021 0.180 -64.340 0.8 4.121 2.069 0.502 -5.985 -62.868 1 4.764 4.082 0.856 -1.350 -57.358 1.5 4.868 3.107 0.638 -3.903 -46.145 2 4.950 1.650 0.333 -9.551 -37.975 4 3.690 1.665 0.451 -6.916 -21.320 6 3.381 1.145 0.338 -9.421 -14.584 8 1.911 0.908 0.475 -6.466 -11.042 16 3.197 0.370 0.115 -18.786 -5.572 32 4.978 0.044 0.008 -41.938 -2.793 64 3.517 0.055 0.015 -36.478 -1.397