A double chain of coupled circuits in analogy with mechanical lattices

. A unitary transformation obtained from group theoretical considerations is applied to the problem of finding the resonant frequencies of a system of coupled LC-circuits. This transformation was previously derived to separate the equations of motion for one dimensional mechanical lattices. Computations are performed in matrix notation. The electrical system is an analog of a pair of coupled linear lattices. After the resonant frequencies have been found, comparisons between the electrical and mechanical systems are noted.


INTRODUCTION.
This brief note is a by-product of work done in mechanics rather than circuit analysis. But, having worked problems involving coupled mechanical oscillators, simple changes of names have provided us with results for linearly coupled circuits since the underlying mathematics of eigenvalues and eigenfunctions is the same for the mechanical and electrical systems. We claim no specific relevance for our work to matters of immediate practical concern to electrical engineers. However, we do submit our paper in hopes that readers concerned with such topics as coupled transmission lines and translationally invariant circuits will find both the analogy to mechanical lattices and the mathematical exercise to be of interest. [1] In our note to follow, we shall apply to a system of coupled LC-circuits a unitary transformation which was derived from the symmetries of a mechanical analog of the electrical system. In the mechanical problem, the transformation separated the equations of motion for a one-dimensional lattice of N identical particles having nearest neighbors coupled with harmonic springs. The geometry of the linear brray of springs and particles was simplified by using the Born cyclic condition to convert the lattice into a circular ring with the equilibrium positions of the masses at the vertices of a regular, plane N-gon. The symmetry group of the linear array then became the rotation group C(N) for which the rotation by radian serves as a generator, and the irreducible matrix representations of C(N) determine the entries of the unitary transformation matrix As we proceed, it should become apparent that U does indeed possess the properties which simplify our calculations. We direct readers interested in the construction of the matrix U to the references cited at the conclusion of this introduction. In our work with mechanical lattices, we wrote a Lagrangian for each system in matrix notation. Then we diagonalized that Lagrangian matrix by performing a similarity transformation with U. From the transformed Lagrangian, the natural frequencies of vibration for the system under consideration were readily obtainable. [2,3,4,5,6,7] 2.
COUPLED CHAINS OF LC-CIRCUITS. Let us consider a linear, double array of LC-circuits with 2N circuits in all. By application of the Born condition, we can connect the first and (2N--1)-th circuits and the 2-nd and 2N-th circuits to obtain the circular array with the connection as indicated in Figure I. We desire to compute the resonant frequencies of this system. We recognize three, interwoven linear arrays of circuits, each array being analogous to a linear lattice of particles and springs: We also recognize that the permutation P which sends circuit k to circuit k + l(Mod 2N) is a symmetry operation for the system and that the group G {P, P2, p3 p2N} is a symmetry group of the double array once we have connected the first and second circuits to the (2N 1)-th and 2N-th circuits. Furthermore, G is isomorphic to the rotation group C(2N).
In Figure 1, there are 2N current loops indicated. In the k-th circuit, dlk denotes the current in its loop while =/ lkdt gives the charge associated with that current on each capacitor. Figure  The technique for solution of this differential equation by letting Pk is well known. We find that, if (R + 4R sin22 + 4R sin --) < 4L( sin2 + sin -), we obtain the resonant frequencies t(k) 3.

OBSERVATIONS.
In the event that R=R1---R 0, the frequency distribution for k {1,2,3 2N} reduces to that for coupled linear lattices for which there is no energy loss during oscillation.
The array of circuits corresponds to a double chain of identical particles as shown in Figure 3. Line segments between vertices indicate connecting ideal springs. The unit cell for the double chain is a parallelogram, and the triangles with vertices k, k+l, k+2 are isosceles. The frequencies computed correspond to longitudinal vibrations which are parallel to the center line (CL) of the chain.
We observe that the chains, 1,3,5 2N-1 and 2,4,6 ,2N, are uncoupled by letting C1 o. If the resistances are all taken to be zero, the resulting frequency distribution is just that for longitudinal vibrations in a linear lattice of N particles with mass numerically equal to L which are connected by harmonic springs of force constant [2].