Properties of Translation Operator and the Solution of the Eigenvalue and Boundary Value Problems of Arbitrary Space-time Periodic Circuits

Abstract

The time periodic circuit theory is exploited to introduce an appropriate translation operator that is invariant under the change of the spatial unit cell. Useful properties of the operator are derived. By casting the problem in an eigenvalue problem form, the equivalency between solutions at different positions along the structure is demonstrated. It is shown that the underlying mathematical machinery is identical to the one used in the analysis of linear time invariant periodic structures, where a two step eigen-decompositions is performed. The first decomposition is in the temporal eigenfunctions basis, which is followed by the decomposition of the translation operator in the spatial domain. The two step process results in the well-known dispersion relation. We also prove that all points in the ($\beta$,$\omega$) plane parallel to the modulation velocity are equivalent in the sense that the eigenvectors are related by a shift operator. Additionally, the wave propagation inside the space time periodic circuit and the terminal characteristics are rigorously determined via the expansion of the total solution in terms of the eigenmodes. To validate the developed framework, two examples are provided. In the first, a space time modulated composite right left handed transmission line is studied and results are compared with time domain simulation. The second example is concerned with the characterization of the non-reciprocal behaviour observed on a nonlinear transmission line that was manufactured in our lab. Using the developed machinery it is shown that the passive interaction between different harmonics results in an observed giant non-reciprocity, where the difference between the forward and backward transmission coefficients can be greater than 30 dB. The frequencies at which non-reciprocity occurs and its strength agree with time domain simulation and measurements.