Name _____________________________________________ Mailbox __________________ 1 ECE-205 Exam 1 Winter 2011 Calculators can only be used for simple calculations. Solving integrals, differential equations, systems of equations, etc. does not count as a simple calculation. You must show your work to receive credit. Problem 1 _________/10 Problem 2 _________/20 Problem 3 _________/30 Problem 4 _________/22 Problems 5-10 ________/18 Total _____________Name _____________________________________________ Mailbox __________________ 2 1) (10 points) For a first order system described by the differential equation (1() ) ( 1)2ty tyt xt−=− with 00t = and 0() 1yt =, use integrating factors to solve the differential equation. Include the initial conditions in your solution.Name _____________________________________________ Mailbox __________________ 3 2) (20 points) Assume we have a first order system with the governing differential equation 0.4 ( ( 3))()yt y tt x=+ The system has the initial value of 1, so (0) 1y =. The input to this system is 010()3101222ttxttt<−<=<−<≤≤ Determine the output of the system in each of the above time intervals. Simplify your final answer as much as possible and box it. Be sure to include the correct initial value in the first interval!Name _____________________________________________ Mailbox __________________ 4 3) (30 points) For the following three differential equations, assume the input is () 4 ()xt ut= (the input is equal to four for time greater than zero), and the initial conditions are(0) (0) 0yy= = Determine the solution to each of the following differential equations and put your final answer in a box. Be sure to use the initial conditions to solve for all unknowns. You must show all your work to receive credit. a) () 3 () 2 () ()ytyt y xtt =++ b) ˙() ()4 4 () 8 ()yt xtyt yt++= c)() () ())4 6( 41yt yt y tt x++ = Name _____________________________________________ Mailbox __________________ 5 4) (22 points) Derive the governing differential equation for the following circuit. You do not need to put the differential equation into standard form.Name _____________________________________________ Mailbox __________________ 6 Problems 5-10, 3 points each, no partial credit (18 points) Problems 5-7 refer the following graph showing the response of a second order system to a step input. 5) The percent overshoot for this system is best estimated as a) 400% b) -400 % c) 300% d) -300 % e) -33% f) 33% 6) The (2%) settling time for this system is best estimated as a) 0.3 ms b) 0.6 ms c) 1.0 ms d) 1.2 ms 7) The static gain for this system is best estimated as a) 1.5 b) 3 c) -1.5 d) -3 00.20.40.60.811.2-5-4-3-2-10123Time (ms) Output y(t)Input x(t)Name _____________________________________________ Mailbox __________________ 7 Problems 8-10 refer the following graph showing the response of a second order system to a step input. 8) The percent overshoot for this system is best estimated as a) 400% b) 250 % c) 200% d) 150 % e) 100 % f) 25% 9) The (2%) settling time for this system is best estimated as a) 1.5 ms b) 2.5 ms c) 4 ms d) 5 ms 10) The static gain for this system is best estimated as a) 1 b) 2 c) 3 d) 4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 500.511.522.53Time (ms) Output y(t)Input x(t)Name _____________________________________________ Mailbox __________________ 8Name _____________________________________________ Mailbox __________________ 9Name _____________________________________________ Mailbox __________________
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