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HW8_ece220_2007_solution

# HW8_ece220_2007_solution - Homework 8(ECE220 Fall 2007 Due...

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Homework 8 (ECE220 - Fall 2007) Due: Wednesday, November 7 by 2PM in the drop box outside 219 Phillips Hall. Reasoning and work must be shown to gain full/partial credit. WRITE YOUR NAME AND NET ID ON ALL PAGES HANDED IN! 1. (37 points) Consider the LTI system described by the RLC circuit shown in the below figure. x ( t ) is the input and y ( t ) is the output. (a) (5 points) Find the differential equation that relates the input to the output. Solution: V L ( t ) = L dI L ( t ) dt I C ( t ) = C dV C ( t ) dt V C ( t ) = y ( t ) V C ( t ) = x ( t ) - RI R ( t ) - L dI L ( t ) dt I R ( t ) = I L ( t ) = I C ( t ) V C ( t ) = x ( t ) - RC dV C ( t ) dt - L d dt C dV C ( t ) dt · y ( t ) = x ( t ) - RC dy ( t ) dt - LC d 2 y ( t ) dt 2 LC d 2 y ( t ) dt 2 + RC dy ( t ) dt + y ( t ) = x ( t ) (b) (10 points) Determine the frequency response of the system. Let R = 1Ω, C = 1 F , and L = 1 H . Solution: Y ( ω ) = H ( ω ) X ( ω ) Y ( ω ) X ( ω ) = H ( ω ) Y ( ω )( LC ( ) 2 + RCjω + 1) = X ( ω ) H ( ω ) = 1 LC ( ) 2 + RCjω +1 H ( ω ) = 1 - ω 2 + +1 (c) (5 points) Determine the output of the system in part a) if the input is x ( t ) = sin ( t ). Solution: X ( ω ) = π j ( δ ( ω - 1) - δ ( ω + 1)) Y ( ω ) = π j 1 - 1+ j +1 δ ( ω - 1) - 1 - 1 - j +1 δ ( ω + 1) · Y ( ω ) = - π ( δ ( ω - 1) + δ ( ω + 1)) y ( t ) = - cos( t ) (d) (5 points) Determine the output of the system in part a) if the input is x ( t ) = cos (2 t +1).

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HW8_ece220_2007_solution - Homework 8(ECE220 Fall 2007 Due...

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