  Using the time periodic ABCD parameters, an expression for the dispersion
relation of space-time modulated structures is obtained. The relation is valid
for general structures even when the spatial granularity is comparable to the
operating and modulation wavelengths. In the limit of infinitesimal unit cell,
the dispersion relation reduces identically to its continuous counterpart. For
homogeneous space-time modulated media, the time periodic circuit approach
allows the extension of the well-known telegraphist's equations. The time
harmonics are coupled together to form an infinite system of coupled
differential equations. At the scattering centres, where the interaction is
mediated via the modulation (pump wave), the telegraphist's equations reduce to
the interaction of two waves only: the signal and idler. The interaction can
then be described using three wave mixing, satisfying the phase matching
condition. It is demonstrated that the time periodic S parameters provide an
alternative and appealing visualization of the modal conversion and the
emergence of non-reciprocity inside the bandgaps.
