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

Thread Starter


Joined Apr 9, 2017
Dear all,

Im a physicist student struggling with the following problem (blue lines denote periodic boundary condition):
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:
Taking time derivative I get:
applying Kirchhoffs Voltage Law:
upload_2017-11-11_14-5-49.pngsumming over closed path

Assuming a harmonic solution I get:
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.
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

Thank you again for your time!