  We have developed a coupled-mode analysis framework for superconducting
travelling-wave parametric amplifiers using the full Telegrapher's equations to
incorporate loss-related behaviour. Our model provides an explanation of
previous experimental observations regarding loss in amplifiers, advantages of
concatenating amplifiers to achieve high gains, and signal gain saturation.
This work can be used to guide the design of amplifiers in terms of the choice
of material systems, transmission line geometry, operating conditions, and pump
strength.
