Name _____________________________________________ Mailbox __________________ ECE-205 Exam 1 Winter 2009 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 _________ Problem 2 _________ Problem 3 _________ Problem 4 _________ Problem 5 _________ Total _____________ 1Name _____________________________________________ Mailbox __________________ 1) (15 points) Derive the governing differential equation for the following first order circuit. You can use any method you want (except for copying). You do not need to put your answer in a standard form. 2Name _____________________________________________ Mailbox __________________ 2) (30 points) Assume we have a first order system with the governing differential equation () 4 ()28yt t ty ()x=+ The system is initially at rest, so(0) 0y=. The input to this system is 01030()21111.5.5ttxttt⎧⎪<⎪=⎨−<<≤≤≤⎪⎪⎩ Determine the output of the system in each of the above time intervals. Simplify your final answer as much as possible and box it. 3Name _____________________________________________ Mailbox __________________ 3) (15 points) Assume we have a first order system with the governing differential equation () 3 () ( 12)tyt yt e xt−+=− The initial time is and initial value01t =(1) 2y=. Use the method of integrating factors to determine the output as a function of the (unknown) input(yt) ()xt. Simplify your answer as much as possible and box it. 4Name _____________________________________________ Mailbox __________________ 4) (20 points) For the second order circuit below, derive the governing second order differential equation for the output and input()yt ()xt. You do not need to put it into a standard form. 5Name _____________________________________________ Mailbox __________________ 5) (20 points) The form of the under damped (01ζ<<) solution to the second order differential equation 22() 2 () ()= K ()nn nyt yt yt xtζω ω ω++ for a step input () ()xtAut= is () sin( )ntdyt KA ce tζωωφ−=++ where c and φ are constants to be determined and the damped frequency 21dnωωζ=− a) Using the initial condition show that (0) 0y =21tan( )ζφζ−= b) We can express the relationship in part a using the following triangle. Fill in the blanks and then use this triangle determine an expression for sin( )φ. c) Use your answer to part b, and the initial condition (0) 0y= to determine the remaining unknown constant, and write out the complete solution for . ()yt 6Name _____________________________________________ Mailbox __________________ 7Name _____________________________________________ Mailbox __________________ 8Name _____________________________________________ Mailbox __________________
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