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Trying to calculate normal modes of nearly infinite network LC circuits

  1. Oct 1, 2010 #1
    1. The problem statement, all variables and given/known data
    The first circuit has a capacitor with capacitance c and an inductor with inductance L. In series with this is another capacitor which is connected to the next loop in the circuit.

    It look something like http://imgur.com/YJDaD.png"
    Sorry for the crude drawing.

    2. Relevant equations
    The first part of the problem was to prove the equation could be written as dI2/dt2=w02(Ii-1-2Ii+Ii+1)

    So that is a relevant equation and I have managed to do that just fine with Kirchoff's laws.


    3. The attempt at a solution
    The second part is to find the normal frequencies. Now I understand that this is very similar to a beaded string problem, or a discrete wave, however in the wave equation there is a Sin(k*xi) and I cannot for the life of me figure out what the equivalent equation is for a circuit.
    I know that the current will repeat both with the individual loops and each loop will repeat with time, but I cannot figure out how to represent the space part of the wave equation in a circuit setting. This will be necessary for me to solve the differential equation to get the normal modes...

    Help?
     
    Last edited by a moderator: Apr 25, 2017
  2. jcsd
  3. Oct 1, 2010 #2

    diazona

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    Homework Helper

    Unless I'm missing something, the equation you have there can be written as
    [tex]\frac{\mathrm{d}^2}{\mathrm{d}t^2}\begin{pmatrix}I_1 \\ I_2 \\ \vdots \\ I_{n-1} \\ I_{n}\end{pmatrix} = \omega_0^2\begin{pmatrix}-2 & 1 & 0 & 0 & \ddots \\ 1 & -2 & 1 & \ddots & 0 \\ 0 & 1 & \ddots & 1 & 0 \\ 0 & \ddots & 1 & -2 & 1 \\ \ddots & 0 & 0 & 1 & -2\end{pmatrix}\begin{pmatrix}I_1 \\ I_2 \\ \vdots \\ I_{n-1} \\ I_{n}\end{pmatrix}[/tex]
    Are you familiar with equations of this type, i.e. would you know how to solve it? It's the same mathematical procedure that is used in the discussion of coupled oscillators.
     
  4. Oct 1, 2010 #3
    I feel like to find the resonant frequency in the matrix fashion I would just take the determinant of the matrix and set it to zero revealing the eigenvalues.

    However to answer your question, no, I wouldn't know how to solve it, but am willing to do some reading if you send me in the correct direction.

    I have taken linear algebra so it shouldn't be too difficult to learn, this material just has not been presented to me in that fashion just yet.
     
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