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# ps3 - EE 350 PROBLEM SET 3 DUE 24 Sep 2007 Reading...

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EE 350 PROBLEM SET 3 DUE: 24 Sep 2007 Reading assignment: Lathi Sections 2.5 and 2.6 Exam I is scheduled for Thursday, September 27 from 8:15 pm to 10:15 pm in room 121 Sparks 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/scientific calculators are capable of graphing functions and solving ODEs: skills that you must be capable of doing by hand. Sections will meet for Laboratory #1 during the week of September 17. Problem 11: (18 points) Consider the first-order passive circuit in Figure 1. Figure 1: First-order passive circuit with input voltage f ( t ) and output voltage y ( t ). 1. (4 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 C . 2. (1 point) State the characteristic equation and find its root(s) in terms of the parameters R 1 , R 2 , and C . 3. (1 point) In general a characteristic root λ that maybe complex. 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 defined as τ = 1 / Re( λ ). The time constant τ is a measure of how quickly a natural mode relaxes towards zero. The smaller

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ps3 - EE 350 PROBLEM SET 3 DUE 24 Sep 2007 Reading...

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