Propagation Characteristics of Finite Lengths of Periodic Structures

In linearized (matrix) models of periodic structures the propagation characteristics, or unforced solutions, are the eigenvectors of the transfer matrix for a single period of the structure. This solution is, however, only applicable to an infinite length of the structure. If n periodic lengths (or sections) are cascaded the overall transfer matrix is the n-th power of the matrix for one periodic length, which can be reduced by means of the Cayley-Hamilton theorem. Examination of this reduced matrix reveals that the propagation characteristics of a finite length of a periodic structure can be expressed in terms of rationalized Tchebychef polynomials. The two analyses are shown to converge as n¿ ¿. An example is presented to illustrate the differences between a finite length and an infinite line.