EE 321 Project Spring 2018

Transcription

1 EE 21 Projet Spring 2018 This ourse projet is intended to be an individual effort projet. The student is required to omplete the work individually, without help from anyone else. (The student may, however, get help from the instrutor or TA.) The student must turn in this page with the preliminary report, and the final report, inluding signature on the pledge at the bottom. The final projet report is due at the beginning of lass leture on Wednesday, April 18. Late reports may be turned in on Friday, April 20 for a 20% redution in grade. Reports submitted after April 20 will be assigned a grade of 0. The projet tasks are outlined on the following pages. The student must produe a typed report, inluding equations, figures, and tables, that inludes disussion of the results and any onlusions reahed. The report may ontain a omparison of tehniques, a disussion of the behavior of the iruit as a filter, and any positive outomes of the efforts of the projet. The guidelines for good tehnial report writing need to be followed. We are partiularly interested in reports demonstrating abilities related to the following ABET Criteria outomes: (a) an ability to apply knowledge of mathematis, siene, and engineering; (b) an ability to design and ondut experiments as well as analyze and interpret data; (g) an ability to ommuniate effetively; (k) an ability to use tehniques, skills, and modern engineering tools neessary for engineering pratie. Noties (standard Shool EE 21 projet requirements): 1) A preliminary report, onsisting of Part I, tasks 1), 2) and ), is due Wednesday, Marh 28, at the beginning of lass leture. For this preliminary due date, turn in only the results, the MATLAB ommands, and a signed opy of the pledge below; no typed report is needed. Your results should present a derivation of the differential equation and the state spae model. This preliminary report is worth 10 points out of the total 100 points on the projet grade. A late preliminary report may submitted up to the next leture period (that is, up until Friday, Marh 0) for a 20% redution in that part of the grade (i.e., for a maximum of 8 points). The final report must also inlude all of Part I. 2) Any evidene of ollaboration with other students on either the final projet report or the preliminary results will result in a grade of zero for the projet for all ollaborators. Additionally, the violation will be reported to the Shool of EECS and the University. ) Write the report in your own words. Grading will be based, in part, on ontent and grammar, so be sure to proof read your report. If signifiant portions of the report are found to be opied from another soure, without proper attribution, a grade of zero will be assigned for the projet and the plagiarism reported to the Shool and University. 4) The pledge below must be signed and turned in with both the preliminary report and the final projet report. Final projet reports without the signed pledge will reeive a grade of zero. PLEDGE: I HAVE NOT OBTAINED ASSISTANCE IN COMPLETING THIS PROJECT FROM ANYONE OTHER THAN THE INSTRUCTOR OR TA FOR THIS COURSE, NOR HAVE I GIVEN ASSISTANCE TO OTHER MEMBERS OF THIS CLASS. SIGNED: 1

2 A iruit with input voltage, v i (t), and output voltage, v ( ), is shown below. vv LL (tt) RR 1 LL (t) v i CC 1 R 2 CC 2 v ( ) Figure 1. Ciruit with input voltage, vv ii (tt), and output voltage, vv 0 (tt). Part Assume allzero initial onditions and use sdomain tehniques to determine HH(ss) = VV 0(ss) VV ii (ss). 2. Using Kirhhoff s voltage and urrents laws, derive a rd order differential equation for vv 0 (tt) = vv CC2 (tt) in the iruit above. Assume that there is no energy stored in the apaitors or indutor at time t = 0. From the differential equation, find a state variable representation, and speify the state variable matries A, B, C, D, to generate the outputs, voltages v ( ) and vv LL(tt).. Let the iruit elements have parameter values LL = ( αα)HH, CC 1 = 1 μμμμ, CC 2 = 250 nnnn, RR 1 = 50 ΩΩ, and RR 2 = 1, αα ΩΩ, where αα = 0. xxxxxx, with xyz the final three digits of your student ID number. The value of 0. xyz satisfies 0 0. xyz 1, so the value of LL lies in the range [0.15, 0.55]HH, and the value of RR 2 lies in the range [1, 5] kkkk. Use Matlab to determine the response of the iruit ( v 0 ( t ) and vv LL(tt)) over a suitable time interval (roughly [0, 10] mse) to input v i ( t) = u( t) V, where u(t) is the unitstep signal. Determine the d gain of the filter, gg, and aurately plot the saled unit step response, vv 0 (tt)/gg. Determine (from the saled numerial response) the 100% rise time, perent overshoot, and 2% settling time. (Note that the default rise time value that some Matlab funtions provide is the 10% to 90% rise time, so make sure your numerial result mathes your plot and orresponds to the 100% rise time.) Verify that v 0 ( t ) and vv LL (tt) have the orret values at tt = 0 and approah the orret steadystate values as t. 2

3 Part II. 4. Using the same omponent values as in task, omplete the following. a. Identify all poles and zeros of HH(ss) and plot their loation in the splane (e.g., using the Matlab pzmap funtion). b. Use the Matlab transfer funtion ommand (e.g., >> sys = tf( ) ) to define a system objet and then use the Matlab lsim( ) ommand with this sys objet to find the system (HH(ss)) unit step response. Compare this to the result for v0 ( t) in task (and they should be idential).. (Use Matlab to) Plot the Bode magnitude and phase frequeny responses for the system. d. (Use Matlab to) Find the response to input vv ii (tt) = 80 os(2,50t) uu(tt). Evaluate the response for a suitable length of time so that the response reahes steady state. Then, determine the magnitude and phase of the output v 0 ( t ), and ompare to the magnitude and phase of your Bode plots at frequeny 2,500 rad/se. Find, by hand alulation, the sinusoidal steadystate response to the input vv ii (tt) = 80 os(2,50t). Plot this response in the same figure as the input and the Matlab omputed response. Answer the following questions: How long does it take for the Matlab omputed response to (approximately) reah the steady state sinusoidal response? How is this time duration related to the poles of the system transfer funtion, HH(ss). Part III 5. For Part III, set the value of resistor RR 2 =, so that it is effetively removed from the iruit. Find the transfer funtion HH(ss) = VV 0(ss) VV ii (ss), and omplete the following. a. Set RR 1 = 1 ΩΩ, LL = 4 HH, CC 1 = 1/2 FF and CC 2 = /2 FF and find all poles of HH(ss). This is a normalized (to utoff frequeny 1 rad/s) rd order Butterworth filter transfer funtion. By hand alulation, determine the system unitstep response, and plot the result. b. Let RR 1 = 50 ΩΩ, and let the utoff frequeny be ωω = 10,000 10,000αα, where again αα =. xxxxxx and xyz are the last three digits of your student ID number. Design the filter to be a rd order Butterworth filter with db utoff frequeny ωω rad/s. This will involve, for your speified value of ωω, properly seleting the values of CC 1, CC 2 and LL. Note that the desired Butterworth filter transfer funtion must have the form H ( s) = ω 2 2 s 2ω s 2ω s ω, where ω is the utoff frequeny. After designing the filter, omplete the following. (Note: Use the expression for HH(ss) above, rather than the numerial values based on your design, in the Part III tasks below.) i. Find the pole loations (e.g., using MATLAB), and plot the pole loations in the splane. Identify the angle of the omplex poles in the splane and verify that they orrespond to a rd order Butterworth filter.

4 ii. Use state variables and MATLAB to determine the unit step response. Aurately plot the step response and determine the 100% rise time, maximum perent overshoot, and 2% settling time. iii. Aurately plot the filter frequeny response (use Bode plots). Verify that the filter response mathes the desired Butterworth filter frequeny response. iv. The steadystate sinusoidal response of the filter is determined by the frequeny response, evaluated at the sinusoid frequeny. Let the iruit input be vv ii (tt) = 120 os (ωω tt). Use Matlab to ompute the filter response for a suitably long duration so that the response reahes (approximately) the steadystate signal. Plot in a single figure both v i (t) and v ( t 0 ), and identify in your figure that the expeted filter steady state response magnitude and phase are realized. Plot the theoretial sinusoidal steady state solution (also in the same figure) and show that v 0 ( t) approahes this as time gets suffiiently large. Verify that the results math your Bode plots. Approximately how long does it take for the (Matlab) response to reah steadystate? Compare this (observed) time to the settling time found in part ii). Part IV. 6. Return to the iruit shown in Figure 1, but with the values of CC 1, CC 2 and LL from your design in Part III, and with RR 2 = RR 1 = 50 ΩΩ. This models the effet on the Butterworth iruit designed in Part III, of a load resistane (of 50 ohms). a. Determine the transfer funtion of the resulting filter and use Matlab to plot the Bode frequeny response. Note that the lowfrequeny ( d ) gain is no longer 0 db. Determine this d gain, and express it in terms of the iruit parameters. With respet to the lowfrequeny gain, define the new utoff frequeny as the frequeny at whih the magnitude frequeny response is db less that the d gain. From the Bode response plot, determine the utoff frequeny for the loaded filter, denote it as ωω, and note that it has hanged from the intended value, ωω. b. This part of the projet is seleted to speifially address the ABET riteria (b) an ability to design and ondut experiments as well as analyze and interpret data. The task is to modify your design from part III to design a lowpass filter that has RR 2 = RR 1 = 50 ΩΩ, but also has the originally intended utoff frequeny of ωω from Part III. For this task, the utoff frequeny is determined with respet to the d gain. The d gain is unhanged from part IV 6a), but by redesigning the values of CC 1, CC 2 and LL, make the utoff frequeny the intended value ωω (instead of ωω ). This task is intended to be addressed experimentally, and by interpreting the data (e.g., Bode plots) generated by your experiments. The resulting filter will no longer be exatly a Butterworth filter, but its frequeny response magnitude should appear to be very similar to a Butterworth lowpass filter with utoff frequeny ωω, but with a d gain that is not unity (and is the same as in Part IV 6a). In your report, inlude a disussion of what approahes you used (e.g., experiments, data interpretation, et.) to solve the design problem, and the designed values of CC 1, CC 2 and LL. Provide a polezero diagram of the transfer funtion for your designed filter, as well as a Bode frequeny plot, verifying that the orret utoff frequeny is ahieved. 4

5 Report Preparation. In your report, present the results of the above tasks along with the disussion mentioned on the first page, and supporting derivations, analysis, design equations, plots, and MATLAB ode. As a ruleofthumb, the report should be suffiiently detailed so that if you were to refer to it in one year, you ould easily follow the derivations, disussion, and results. However, avoid exessive detail on the derivation of equations or derivation of a solution to a partiular task. Sine the report must be typeset, selet a suitable word proessing system for your work. The LaTeX system is available for free download and is exellent, but requires some learning. Note: It is ommon pratie for students to use this projet report as part of their written work submitted to satisfy the University s Writing Portfolio requirement. So, keep in mind that the report might, later, be evaluated for that purpose. 5