ps3_08 - EE 350 PROBLEM SET 3 DUE 16 September 2008 Reading...

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EE 350 PROBLEM SET 3 DUE: 16 September 2008 Reading assignment: Lathi Sections 2.5 and 2.6 Exam I is scheduled for Thursday, September 25 from 8:15 pm to 10:15 pm in room 102 Forum for all sections. The exam covers material problem sets 1 through 3, and laboratory #1. The exam is closed-book, but you may bring one 8 1/2 by 11 inch note sheet, Calculators are not allowed as graphical/scienti±c calculators are capable of graphing functions and solving ODEs: skills that you must be capable of doing by hand. Recitation #3 will be held this week. Problem 13: (16 points) Consider the ±rst-order passive circuit in Figure 1. Figure 1: RL circuit with input voltage f ( t ) and output current y ( t ). 1. (3 points) A standard form for expressing an ODE is d n y dt n + a n - 1 d n - 1 y dt n - 1 + ··· + a 0 y = b m d m f dt m + b m - 1 d m - 1 f dt m - 1 + + b 0 f. By introducing the derivative operator D d/dt and the polynomials Q ( D )= D n + a n - 1 D n - 1 + + a 1 D + a 0 P ( D b m D m + b m - 1 D n - 1 + + b 1 D + b 0 , an ODE may also be represented by the compact form Q ( D ) y ( t P ( D ) f ( t ) . (1) For the circuit in Figure 1, carefully derive an expression for the ODE that relates the output voltage y ( t ) to the input voltage f ( t ) and express your answer using the form in equation 1. Provide expressions for the polynomials Q ( D ) and P ( D ) in terms of the parameters R 1 , R 2 , and L . 2. (1 point) State the characteristic equation and ±nd its root(s) in terms of the parameters R 1 , R 2 , and L . 3. (1 point) A characteristic root λ maybe real or complex valued. If the real part of λ is strictly negative, then the corresponding natural (or characteristic) mode e λt exponentially decays to zero as t increases. In this case the time constant associated with the natural mode is de±ned as τ = - 1 / Re( λ ). The time constant τ is a measure of how quickly a natural mode relaxes towards zero. The smaller τ is, the faster the mode decays. Specify the time constant τ for the circuit in Figure 1 terms of the parameters R 1 , R 2 , and L . 4. (3 points) Express the zero-state unit-step response y ( t ) for t 0 in the form y ( t Ae - t/τ + B, and specify the parameters A and B in terms of the parameters R 1 , R 2 , and L . 5. (4 points) For a unit-step input, the time required for the response to increase from 10% to 90% of its ±nal value is de±ned as the rise-time . Show that the rise-time for the ±rst-order system considered in this problem is t r = τ ln 9 . This is a general result that holds for any ±rst-order system when P ( D b o (or equivalently, m = 0) and a characteristic root that is strictly negative.
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6. (4 points) For a unit-step input, the time t s required for the zero-state response to reach and stay within 1% of its ±nal value is called the settling time . Show that for the ±rst-order system the settling time is given by t s = τ ln 100 . This is a general result that holds for any ±rst-order system with m = 0 and a characteristic root that is strictly negative.
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This note was uploaded on 10/13/2008 for the course EE 350 taught by Professor Schiano,jeffreyldas,arnab during the Fall '07 term at Penn State.

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ps3_08 - EE 350 PROBLEM SET 3 DUE 16 September 2008 Reading...

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