I am looking for normal modes of rectangular LC lettuce n x n

Discussion in 'Homework Help' started by Partyzoa, Nov 11, 2017.

  1. Partyzoa

    Thread Starter New Member

    Apr 9, 2017
    3
    0
    Dear all,

    Im a physicist student struggling with the following problem (blue lines denote periodic boundary condition):
    (http://tinyurl.com/yb9s9jqy)
    upload_2017-11-11_13-52-28.png
    It is the easiest of all the n x n possibilities for which I must find all normal modes.

    In one of (I think) best approaches I took currents through each LC as a representative variables to describe the system.

    With assumption that capacitor has the same but opposite amount of charge on each plate I write:
    upload_2017-11-11_13-56-58.png
    Taking time derivative I get:
    upload_2017-11-11_13-58-6.png
    applying Kirchhoffs Voltage Law:
    upload_2017-11-11_14-5-49.png summing over closed path

    Assuming a harmonic solution I get:
    upload_2017-11-11_14-6-52.png
    Now I tried to find linearly independent set of equations:
    I took currents flowing from left to right and from top to the bottom as a positive.
    upload_2017-11-11_14-8-11.png
    First three are closed loops around cells, next two loops around boundary and last three just the current conservation at each nod.

    With calculating a determinant of the system I found angular frequency (maybe I can not justify that) but I do not know how to continue with normal modes as this is not an eigen vector probelm.

    I would greatly appreciate help. Even just a hint. This is a last subject for me to graduate. I asked quite a few PhD colleagues and we all struggled so I have a filling this might not be as trivial as professor think:D

    This is a one solution I found using online simulator
    http://tinyurl.com/yb9s9jqy

    Thank you again for your time!
     
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