ECE486 - Lab1 Report

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Mechanical Engineering

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Jan 9, 2024

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Lab 1 S IMULATION U SING THE A NALOG C OMPUTER 1 of 8 Report By: Justin Habana Lab Partner: John Truong Lab TA: Usman Ahmed Syed Section: AB2 Question 1. ___/15 Theoretical/Experimental Results ___/5  M p Theory % M p Expmt % t r Theory (s) t r Expmt (s) t s Theory (s) t s Expmt (s) 2.0 0.0000 0.0000 10.7000 8.1000 11.6000 10.8400 1.5 0.0000 0.0000 6.3750 6.0800 8.3000 8.2000 1.0 0.0000 0.0000 3.3500 3.4400 5.0000 4.6800 0.8 1.5165 0.9533 2.5040 2.5800 3.6800 3.4200 0.7 4.5988 4.0604 2.1590 2.1200 3.0200 2.7600 0.5 16.3034 15.0568 1.6250 1.6400 6.2791 5.0600 0.3 37.2326 36.9014 1.2990 1.3000 10.1430 7.9400 0.2 52.6621 53.2074 1.2140 1.2000 15.0807 13.7000 Table 1: Theoretical/Experimental Results Figure 1: Response from system w/ 𝜁 = 0.2 Comparison of Theoretical vs. Experimental Results ___/5 The percentage error for percent peak overshoot, rise time and setting time are shown in Table 2. We see that most measured values are within percent 10% error. Some reasons for the deviation include the presence of resistance, noise and imprecision in the physical devices and components used in the experiment. Another depiction of this deviation is shown in Figure 2. In which, rise time at 𝜁 = 2.0 contains the highest deviation between theoretical and observed values. Total: ____/40

Lab 1 S IMULATION U SING THE A NALOG C OMPUTER 2 of 8  M p %𝑒???? t r %𝑒???? t s %𝑒???? 2.0 0.000 32.0988 7.0111 1.5 0.000 4.8520 1.2195 1.0 0.000 2.6163 6.8376 0.8 59.0768 2.9457 7.6023 0.7 13.2586 1.8396 9.4203 0.5 8.2789 0.9146 24.0938 0.3 0.8975 0.0769 27.7451 0.2 1.0249 1.1667 10.0782 Table 2: Percent error between Theoretical vs. Experimental Results Discussion of variation with  of M p , t s , t r ___/5 Figure 2: Relationship between 𝜁 and 𝑀 𝑝 , ? ? and ? ? We observe the following: which are consistent with the trends found in the theoretical calculations. • 𝑀 𝑝 decreased as 𝜁 increased. This increase appears exponential. • ? ? increased as 𝜁 increased. • ? ? decreased as 𝜁 increased, until 𝜁 = 0.7 where ? ? began to increase as 𝜁 increased.

Lab 1 S IMULATION U SING THE A NALOG C OMPUTER 3 of 8 Question 2. ___/15 Effect of  on Pole Locations (Derive Equation and Explain) ___/5 The transfer function of the system is … 𝐻(?) = 1 ? 2 + 2𝜁? + 1 Its poles are … ? = −𝜁 ± 𝑗√1 − 𝜁 2 These poles are plotted in Figure 3. Figure 3: Scatter of Pole Locations. Data point labels represent values of 𝜁 . Effect of Pole Locations for an Underdamped System ___/5 In underdamped system has 𝜁 < 1 As  increases, the poles approach ? = −1 . Focusing only on 𝜁 < 1 in Figure 2. We see that as 𝜁 increases … - M p decreases exponentially. - t r increases at an increasing rate. - t s deceases until 𝜁 = 0.7 . From 𝜁 = 0.7 onward, ? ? increases. Effect of Pole Locations on M p , t s , t r for an Overdamped/Critically Damped System ___/5 An over-damped system has 𝜁 > 1 A critically damped system has 𝜁 = 1 As  increases, the poles move away from ? = −1 but remain real. Focusing only on 𝜁 ≥ 1 in Figure 2. We see that as 𝜁 increases … - M p is zero. - t r increases. - t s increases.

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Lab 1 S IMULATION U SING THE A NALOG C OMPUTER 4 of 8 Question 3. ___/10 1 2 min 1 2 1 2 min ( ) ( ) ( )( ) p p p H s H s s p s p s p =  = + + + Similarities/Differences on Overdamped 2 nd -Order system to a 1 st - Order System with the less negative of the 2 nd - Order’s poles ___/6 First, we solve for ? 1 and ? 2 in terms of 𝜁 a 2 nd order system. 𝐻 1 (?) = ? 1 ? 2 (? + ? 1 )(? + ? 2 ) = ? 1 ? 2 ? 2 + ?(? 1 + ? 2 ) + ? 1 ? 2 ? 1 ? 2 = 𝜔 ? = 1 ? 1 + ? 2 = 2𝜁 ? 1 + 1 ? 1 = 2𝜁 ? 1 2 + 1 = 2𝜁? 1 ? 1 2 − 2𝜁? 1 + 1 = 0 2𝜁 ± √4 𝜁 − 4 2 = 2𝜁 ± 2√𝜁 2 − 1 2 = 𝜁 ± √𝜁 2 − 1 = 𝜁 ± √(−1)(1 − 𝜁 2 ) = 𝜁 ± 𝑗√1 − 𝜁 2 𝜁 ? 1 = 𝜁 ± 𝑗√1 − 𝜁 2 ? 2 = 1/? 1 1.5 0.3820, 2.6180 2.6180, 0.3820 5.0 0.1010, 9.8990 9.899, 0.1010 40.0 0.0125, 79.9875 79.9875, 0.0125 Noticing the symmetry in the results from the previous table, we choose only one value for ? 1 and ? 2 . 𝜁 ? 1 = 𝜁 + 𝑗√1 − 𝜁 2 ? 2 = 1/? 1 𝑃 ?𝑖? = min(? 1 , ? 2 ) 1.5 0.3820 2.6180 0.3820 5.0 0.1010 9.899 0.1010 40.0 0.0125 79.9875 0.0125 Solving the prototype 2 nd order system step response at 𝜔 ? = 1 results in … 𝑦(?) = 1 − 𝑒 −𝜁? (cos (?√1 − 𝜁 2 ) + 𝜁 √1 − 𝜁 2 sin (?√1 − 𝜁 2 )) Solving the 1 st order system step response results in … 𝑦(?) = ℒ { 1 ? ? ?𝑖? ? + ? ?𝑖? } = ℒ { ? ? + ? ? + ? ?𝑖? } = ℒ { 1 ? − 1 ? + ? ?𝑖? } = 1 − 𝑒 −𝑝 ?𝑖? ?

Lab 1 S IMULATION U SING THE A NALOG C OMPUTER 5 of 8 Figure 4: Unit Response for 2 nd and 1 st order systems of similar properties (when both plots merge, only the color for the 1 st order system is shown) From Figure 4, we observe the following … • Both 2 nd and 1 st order system responses follow the same general plot and trend. • Higher 𝜁 leads to better 1 st order system approximations of the 2 nd order system. • Both 2 nd and 1 st order system responses lead to the same steady state value. • As ? increases, the 1 st order system continuously became a better approximation of the 2 nd order system. Effect of magnitude of  on the accuracy of the approximations ___/4 Higher 𝜁 leads to better 1 st order system approximations of the 2 nd order system. This is due to the oscillatory term in the 2 nd order system response becoming smaller as 𝜁 increases. Attachments (3) • Plots obtained during lab • Sample response with relevant points for calculating M p , t s and t r marked • Step Responses comparing 2 nd order systems and 1 st order approximations.

Lab 1 S IMULATION U SING THE A NALOG C OMPUTER 6 of 8 Attachment 1: Plots obtained during lab.

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Lab 1 S IMULATION U SING THE A NALOG C OMPUTER 7 of 8 Attachment 2: Sample response with relevant points for calculating Mp, ts and tr marked. Response from system w/ 𝜁 = 0.2

Lab 1 S IMULATION U SING THE A NALOG C OMPUTER 8 of 8 Attachment 3: Step Responses comparing 2 nd order systems and 1 st order approximations. Unit Response for 2 nd and 1 st order systems of similar properties (when both plots merge, only the color for the 1 st order system is shown)