20095ee102_1_EE102_Hw3_solution

20095ee102_1_EE102_Hw3_solution - a) Find the input-output...

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Fall 2009 EE102 HW #3 Due October 21 1. Find the Laplace transforms of the following functions: a) t e t t x 3 2 ) ( = 0 , t 2 2 ( ) ( 3) X s s = - b) ) cos( ) ( wt t x = 0 , t 2 2 ( ) s X s s w = + 2. Using Laplace transform for the given differential equation: ) ( ) ( 4 ) ( 2 2 t u t x t x dt d = + a) Find the Transfer function and its inverse transform. 2 2 ( ) 4 ( ) ( ) ( ) 1 1 ( ) ( ) sin(2 ) ( ) 4 2 s X s X S U s X s H s h t t U s s + = = = = + b) Assume u(t) = 0, and find the zero-input response 2 2 2 ( ) (0) '(0) 4 ( ) 0 (0) '(0) 1 ( ) ( ) (0)cos(2 ) '(0)sin(2 ) 4 4 2 s X s sx x X S sx x X s x t x t x t s s - - + = = + = + + + c) Now assume the input u(t) is the unit step response: 0 , 1 ) ( = t t u Find the output x(t) given the following initial conditions: 0 ) 0 ( , 2 ) 0 ( ' = = x x
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2 2 2 2 2 2 1 ( ) (0) '(0) 4 ( ) 1 (0) '(0) 1 4 (1 4) 2 1 1 ( ) ( ) ( ) cos(2 ) sin(2 ) ( 4) 4 4 4 4 4 4 s X s sx x X S s sx x s X s x t u t t t s s s s s s s - - + = = + + = - + = - + + + + + + 3. For the given circuit below, The input is u(t) and output is v(t) (voltage across the capacitor). R = 400 Ω, L = 100 H, C = 5 mF
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Unformatted text preview: a) Find the input-output relation as a differential equation. 2 2 2 2 ( ) ( ) ( ) ( ) ( ) 4 ( ) 2 ( ) 2 ( ) d R d v t u t v t v t dt L dt LC LC d d v t v t v t u t dt dt + + = + + = b) Find the Transfer function and its inverse transform. 2 2 (2 2) (2 2) ( ) 4 ( ) 2 ( ) 2 ( ) ( ) 2 2 1 2 1 2 ( ) ( ) 4 2 ( 2 2)( 2 2) ( 2 2) ( 2 2) 1 1 ( ) 2 2 t t s V s sV s V s U s V s H s U s s s s s s s h t e e-+--+ + =-= = = = + + + + + + -+ + + -- = + c) Calculate the gain and phase of the Transfer function. 2 2 2 2 1 2 2 ( ) 4 2 2 ( ) , (2 ) 16 4 ( ) tan ( ) 2 s jw H s w j w H jw w w w H jw w =-=-+ + =-+ = --R...
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This note was uploaded on 01/16/2012 for the course EE 102 taught by Professor Levan during the Spring '08 term at UCLA.

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20095ee102_1_EE102_Hw3_solution - a) Find the input-output...

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