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# lec09-handout-6up - 6.003 Signals and Systems Lecture 9...

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6.003: Signals and Systems Lecture 9 October 8, 2009 1 6.003: Signals and Systems Second-Order Systems October 8, 2009 Last Time We analyzed a mass and spring system. x ( t ) y ( t ) F = K ( x ( t ) y ( t ) ) = M ¨ y ( t ) + K M A A 1 x ( t ) y ( t ) ˙ y ( t ) ¨ y ( t ) Y X = K M A 2 1 + K M A 2 Last Time We also analyzed a leaky tanks system. r 0 ( t ) r 1 ( t ) r 2 ( t ) h 1 ( t ) h 2 ( t ) τ 1 ˙ r 1 ( t ) = r 0 ( t ) r 1 ( t ) τ 2 ˙ r 2 ( t ) = r 1 ( t ) r 2 ( t ) + 1 τ 1 A + 1 τ 2 A r 0 ( t ) r 2 ( t ) ˙ r 1 ( t ) r 1 ( t ) ˙ r 2 ( t ) R 2 R 0 = A 1 1 + A 1 × A 2 1 + A 2 Second-Order Systems Today: Look more carefully at growth and decay of oscillatory re- sponses by studying an analogous electrical circuit. v i v o R L C But First ... The canonical forms for CT and DT differ. + A s 0 X Y + Delay z 0 X Y H = Y X = A 1 s 0 A H = Y X = 1 1 z 0 R h ( t ) = e s 0 t u ( t ) h [ n ] = z n 0 u [ n ] A → 1 s R → 1 z s -plane s -plane z -plane z -plane Check Yourself What if we had used the DT canonical form for CT? + A s 0 X Y What is the impulse response of this system? 1. s 0 e s 0 t u ( t ) 2. s t 0 u ( t ) 3. 1 + s 0 e s 0 t u ( t ) 4. δ ( t ) + s 0 e s 0 t u ( t ) 5. none of the above

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6.003: Signals and Systems Lecture 9 October 8, 2009 2 Second-Order Systems Today: Look more carefully at growth and decay of oscillatory re- sponses by studying an analogous electrical circuit. v i v o R L C Second-Order Systems Solve with state variable approach. State variables represent the minimum knowledge of the past ( t < t 0 ) needed to propagate the output into the future ( t > t 0 ). Check Yourself v C i C v L i L i C = C dv C dt v L = L di L dt Which of the following can be state variables? 1. v C and v L 2. i C and v L 3. i C and i L 4. v C and i L 5. i C and v C and i L and v L 6. none of above Second-Order Systems State variable approach: determine expressions for derivatives of state variables in terms of (undifferentiated) state variables.
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lec09-handout-6up - 6.003 Signals and Systems Lecture 9...

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