  We analyze the effect of circuit parameter variation on the performance of
Josephson traveling-wave parametric amplifiers (JTWPAs). Specifically, the
JTWPA concept we investigate is using flux-biased nonhysteretic rf-SQUIDs in a
transmission line configuration, which harnesses the three-wave mixing (3WM)
regime. Dispersion engineering enables phasematching to achieve power gain of
$\sim$20 dB, while suppressing the generation of unwanted mixing processes. Two
dispersion engineering concepts using a 3WM-JTWPA circuit model, i.e., resonant
phase-matching (RPM) and periodic capacitance modulation (PCM), are discussed,
with results potentially also applicable to four-wave-mixing (4WM) JTWPAs. We
propose suitable circuit parameter sets and evaluate amplifier performance with
and without circuit parameter variance using transient circuit simulations.
This approach inherently takes into account microwave reflections, unwanted
mixing products, imperfect phasematching, pump depletion, etc. In the case of
RPM the resonance frequency spread is critical, while PCM is much less
sensitive to parameter spread. We discuss degrees of freedom to make the JTWPA
circuits more tolerant to parameter spread. Finally, our analysis shows that
the flux-bias point where rf-SQUIDs exhibit Kerr-free nonlinearity is close to
the sweet spot regarding critical current spread.
