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Unformatted text preview: Homework 5 (ECE220 - Fall 2007) • Due: Thursday, October 18 at the beginning of lecture. • Reasoning and work must be shown to gain full/partial credit. • WRITE YOUR NAME AND NET ID ON ALL PAGES HANDED IN! 1. (50 points) RC Filter Consider the following resistor-capacitor filter: Let the source voltage V S ( t ) be the input of the system x ( t ) (Equation 1). Let the voltage across the capacitor V C ( t ) be the output of the system y ( t ) (Equation 2). The voltage across the resistor V R ( t ) is related the the current through it i R ( t ) by the Equation 3, while the voltage across the capacitor is related by Equation 4. Via Kirchoff’s Laws we can see that the resistor current must equal the capacitor current (Equation 5). and the source voltage must equal the sum of the capacitor and resistor voltages (Equation 6). V S ( t ) = x ( t ) (1) V C ( t ) = y ( t ) (2) V R ( t ) = i R ( t ) R (3) C dV C ( t ) dt = i C ( t ) (4) i R ( t ) = i C ( t ) (5) V S ( t ) = V R ( t ) + V C ( t ) (6) (a) (5 points) Show that the system is a first order differential equation. Solution: V R ( t ) R = i R ( t ) = i C ( t ) = C dV C ( t ) dt RC dV C ( t ) dt = V S ( t )- V C ( t ) RC dV C ( t ) dt + V C ( t ) = V S ( t ) (b) (5 points) Draw the block diagram of it. Solution: An example way to have drawn it is: (c) (10 points) Find the impulse response assuming that V C (0) = 0. Solution: I am solving this like you would solve any general differential equation. Often times we make simplifications about our signals so that we do not have to solve in this detail for signal problems. I am solving it in full so that I can use some of the results in later parts....
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- Fall '05
- LTI system theory, Impulse response, ej