{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Assignment 3-Solutions

Assignment 3-Solutions - ECE3085 Homework#1 Due Jan 28 2009...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
ECE3085 - Homework #1 Due: Jan. 28, 2009 Problem 1. (5 pts) Compute the inverse Laplace transform of F ( s ) = 1 s ( s + 2) 2 Solution 1. The solution will be obtained by doing the partial fraction expansion of the Laplace domain function F ( s ) then regrouping the numerator terms, F ( s ) = A 1 s + B 11 s + B 12 ( s + 2) 2 = A 1 ( s 2 + 4 s + 4) + B 11 s 2 + B 12 s s ( s + 2) 2 = ( A 1 + B 11 ) s 2 + (4 A 1 + B 12 ) + 4 A 1 s ( s + 2) 2 . Comparing the symbolic partial fraction expansion of F ( s ) to the true F ( s ) leads to the following system of equations 0 = A 1 + B 11 0 = 4 A 1 + B 12 1 = 4 A 1 , whose solution is A 1 = 1 / 4 , B 11 = - 1 / 4 , and B 12 = - 1 . Thus, F ( s ) = 1 4 s + - 1 4 s - 1 ( s + 2) 2 = 1 4 s - s + 4 4( s + 2) 2 = 1 4 s - s + 2 + 2 4( s + 2) 2 = 1 4 s - 1 4( s + 2) - 1 2( s + 2) 2 . Note that you could have done it the other way described in class to arrive at the final form I have above, A 1 = sF ( s ) | s =0 = 1 4 A 3 = ( s + 2) 2 F ( s ) vextendsingle vextendsingle s = - 2 = - 1 2 A 2 = d d s vextendsingle vextendsingle vextendsingle vextendsingle s = - 2 ( ( s + 2) 2 F ( s ) ) = - 1 4 . Either way the inverse Laplace transform is 1 4 1( t ) - 1 4 e - 2 t - 1 2 te - 2 t = 1 4 parenleftbigg 1 - e - 2 t - 2 1 2 te - 2 t parenrightbigg 1( t ) . It is acceptable with or without the 1( t ) factor. Problem 2. (5 pts) The DC gain of a system is the steady-state response of the system to a unit step input. In more mathematical terms, that would mean the limit of the time response as t goes to infinity. What is the the DC gain of the system Y ( s ) = G ( s ) U ( s ) = s + 7 ( s + 1)( s 2 + 4) U ( s ) Solution 2. Oh my! As the discussion in class went, this problem was ill-posed. The imaginary poles imply that the system does not have a steady-state response. That means it gets thrown out and everyone gets full credit. Those who stated that it was ill-posed, get a couple points to boot.
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Problem 3. (10 pts) Solve for the following differential equation using the Laplace transform ¨ y + y = t, y (0) = 1 , ˙ y (0) = - 1 . Solution 3. First step is to transform the equation into the Laplace domain, s 2 Y ( s ) - s ˙ y (0) - y (0) + Y ( s ) = 1 s 2 s 2 Y ( s ) - s + 1 + Y ( s ) = 1 s 2 ( s 2 + 1) Y ( s ) = 1 s 2 + s - 1 s 2 ( s 2 + 1) Y ( s ) = 1 + s 3 - s 2 Y ( s ) = s 3 - s 2 +1 s 2 ( s 2 +1) Looking at the equation, it can be simplified further, Y ( s ) = s 3 - 2 s 2 + s 2 + 1 s 2 ( s 2 + 1) = 1 s 2 + s 2 ( s - 2) s 2 ( s 2 + 1) = 1 s 2 + s s 2 + 1 - 2 1 s 2 + 1 .
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern