We have developed a coupled-mode analysis framework for superconducting travelling-wave parametric amplifiers using the full Telegraphers 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.
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