  Superconducting traveling-wave parametric amplifiers are a promising
amplification technology suitable for applications in submillimeter astronomy.
Their implementation relies on the use of Floquet transmission lines in order
to create strong stopbands to suppress undesired harmonics. In the design
process, amplitude equations are used to predict their gain, operation
frequency, and bandwidth. However, usual amplitude equations do not take into
account the real and imaginary parts of the dispersion and characteristic
impedance that results from the use of Floquet lines, hindering reliable
design. In order to overcome this limitation, we have used the multiple-scales
method to include those effects. We demonstrate that complex dispersion and
characteristic impedance have a stark effect on the transmission line's gain,
even suppressing it completely in certain cases. The equations presented here
can, thus, guide to a better design and understanding of the properties of this
kind of amplifiers.
