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RLC circuit - Differential equation - Stationary current calculation

  1. Jan 20, 2013 #1
    1. The problem statement, all variables and given/known data

    I'm decent with differential equations for RLC circuits, and I KNOW the stationary current will be zero, but I need help to work it out mathematically, because my maths gets me -2.4A (...)

    L: 0.05H
    C: 0.04F
    R: 3 Ω
    U (source) = 1 V

    I've got the transient solution: c1e-10t + C2e-50t

    Known:

    i(0+) = 0 (coil prevents instant change in current)
    i'(0+) = u/L = 20

    Once the capacitor is charged up, it will block the current. In other words, the stationary current will be zero.

    3. The attempt at a solution

    So I could simply declare that, on electrical reasoning, but I want to calculate it. The problem is, my calculation is wrong. Something is wrong.

    Been banging my head against this for a while, hope somebody can help me out!

    Differential equation: L*i'' + R*i' +1/c*i = u'

    i''=0, because i'=20

    u'=0, because u=1

    I call the stationary current A:

    L*0 + R*20 + 1/c A = 0

    A=-20*R*C= -2.4

    What have I done wrong? If I lack information, please let me know.
     
    Last edited: Jan 20, 2013
  2. jcsd
  3. Jan 20, 2013 #2

    vela

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    i'' isn't 0 because i'=20. i'=20 only when t=0.

    You have two approaches: 1. Solve the differential equation and then take the limit as t goes to infinity; 2. Argue what i'' and i' equal when the system has reached steady state, plug those values into the differential equation, and then solve for the current.
     
    Last edited: Jan 20, 2013
  4. Jan 20, 2013 #3
    Hmm.. well going with approach 2, that makes sense. There will be no change of i when the system is stable.

    It's real late here, gonna take a look at it tomorrow. Thanks! I will ask again if I am still stumped heh.

    Edit: And yeah, my assumption about i''= 0 because i'(0)=20 is of course flawed.. It's the Christmas break that made me forget some things!
     
  5. Jan 21, 2013 #4
    Alright, looking at the work again today.

    Well it seems to me I already have the starting values I needed!

    i(0+)=0
    i'(0+)=U/L=20

    I don't need i'' for that aspect.

    And yeah, if I go with the second approach you outlined, using reason and knowledge of the RLC-circuit, there is no change of i when it is stable, so that means the i''(∞)=0.

    Not sure if using infinity in an argument is sound, but it works!
     
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