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RLC Circuit Second Order Differential and Laplace

  1. Jul 10, 2010 #1
    1. The problem statement, all variables and given/known data
    Derive the second order differential equation relating x(t) and y(t).
    Using the Laplace transform, find the total response as a function of the zero input response and the zero state response in the following form.


    2. Relevant equations

    Y(s)=Yzs(s) + Yzi(s)

    3. The attempt at a solution

    Loop1: Rs*i1 + 1/c integral(i1 + i2) dt = xs(t)

    Loop2: 1/c integral(i1 + i2) dtau + Rload*i2 + L di2/dt = 0

    Take derivatives

    Loop1: R di1/dt + (i1 + i2)/C = dx/dt

    Loop2: R di2/dt + L di2/dt + (i1 + i2)/C = 0
     

    Attached Files:

  2. jcsd
  3. Jul 11, 2010 #2

    vela

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    Try solving for i1 in the second equation and substitute into the first equation. Then use y(t)=Rloadi2 to rewrite the equation in terms of y(t).
     
  4. Jul 11, 2010 #3
    Loop2: 1/c integral(i1 + i2) dtau + Rload*i2 + L di2/dt = 0

    I believe there is an error here. If the current through the capacitor is i1 + i2, then i2 is moving counterclockwise, and we know as the current approaches the load resistor, it is approaching the negative end of y(t). Therefore, Rload*i2 should be negative. I haven't checked everything, so there could be more problems.
     
  5. Jul 11, 2010 #4

    vela

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    I think that equation is fine; it's correct if you assume i2 goes in the counterclockwise direction. But I should have said to use y(t)=-Rloadi2. I forgot the OP used the opposite direction than usual on i2.
     
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