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Matlab - ODE, find roots of the characteristic equation for the natural response

  1. Sep 4, 2009 #1
    1. The problem statement, all variables and given/known data

    I need to use MATLAB to solve these problems.

    http://users.bigpond.net.au/exidez/IVDP.jpg

    2. Relevant equations

    MATLAB

    3. The attempt at a solution

    a)

    R1=3.6;
    R2=R1;
    C1=33*10^-6;
    C2=22*10^-6;

    % defining the polynomial constants
    Vs=[R1*R2*C1*C2 (2*R2*C2)+(R1*C1) 1];

    'the roots of the equation are';
    roots(Vs)

    ans =

    -25253
    -4209


    b) Largest time constant:

    numf=1;
    denf=Vs;

    %Converting to Partial fraction to find the inverse Laplace transform

    [r,p,k]=residue(numf,denf)

    r =

    1.0e+003 *

    -5.0505
    5.0505


    p =

    1.0e+004 *

    -2.5253
    -0.4209


    Transfer function:

    [tex]\frac{-5051}{s + 25253} + \frac{5051}{s + 4209}[/tex]


    Based on the inverse Laplace of the partial fractions, the eqn then looks like below:

    Vo(t) = r(2)*exp(p(2)*t) +r(1)*exp(p(1)*t)

    Vo(t) = 5051e^(-4209t)-5051e^(-25253t)

    I believe that is my natural response????

    c) obtain and plot the natural response function versus time (use the first 1ms) for an initial constant Vs(t=0) = 5V

    I have no idea, this is all new to me. The initial constant has me fooled and i dont know how to do it with the roots from part a. Can someone help me out with part c please
     
    Last edited by a moderator: Apr 24, 2017
  2. jcsd
  3. Sep 4, 2009 #2
    example:-
     

    Attached Files:

  4. Sep 7, 2009 #3
    that still doesn't help me much...

    so i know the natural response in S domain. I can find the inverse Laplace to find what it is in time domain so i can plot it against t. but the initial constant Vs(t=0) = 5V still confuses me. Also, this doesnt use the root directly obtained in part a!

    Do i just modify the graph so it starts at 5V ????????
     
    Last edited: Sep 7, 2009
  5. Sep 7, 2009 #4
    The question seems to ask you to set Vs(t) = 5v.
     
  6. Sep 8, 2009 #5
    well this is my understanding of it...

    basically i assume a natural response in the form y= Ce^(at)

    given the roots:

    y= C1e^(-25253t) + C2e^(-4209t)

    i am given Vs(0)= 5V

    however to solve for the constants C1 and C2 dont i need the value of the derivative of Vs??
     
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