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The time periodic circuit theory is exploited to introduce an appropriatetranslation operator that is invariant under the change of the spatial unitcell. Useful properties of the operator are derived. By casting the problem inan eigenvalue problem form, the equivalency between solutions at differentpositions along the structure is demonstrated. It is shown that the underlyingmathematical machinery is identical to the one used in the analysis of lineartime invariant periodic structures, where a two step eigen-decompositions isperformed. The first decomposition is in the temporal eigenfunctions basis,which is followed by the decomposition of the translation operator in thespatial domain. The two step process results in the well-known dispersionrelation. We also prove that all points in the ($\beta$,$\omega$) planeparallel to the modulation velocity are equivalent in the sense that theeigenvectors are related by a shift operator. Additionally, the wavepropagation inside the space time periodic circuit and the terminalcharacteristics are rigorously determined via the expansion of the totalsolution in terms of the eigenmodes. To validate the developed framework, twoexamples are provided. In the first, a space time modulated composite rightleft handed transmission line is studied and results are compared with timedomain simulation. The second example is concerned with the characterization ofthe non-reciprocal behaviour observed on a nonlinear transmission line that wasmanufactured in our lab. Using the developed machinery it is shown that thepassive interaction between different harmonics results in an observed giantnon-reciprocity, where the difference between the forward and backwardtransmission coefficients can be greater than 30 dB. The frequencies at whichnon-reciprocity occurs and its strength agree with time domain simulation andmeasurements.

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