  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.
