The unique nature of this transmission line is that its nonlinearity can be tuned with an external magnetic flux and can even change sign. This feature of the transmission line can be used to perform phase matching in a degenerate four-wave mixing process which can be utilized for parametric amplification of a weak signal in the presence of a strong pump. Numerical simulations of the TWPA design have shown that with tuning, phase matching can be achieved and an exponential gain as a function of the transmission line length can be realized. The flexibility of the proposed design can realize: compact TWPAs with less than 211 unit cells, signal gains greater than 20 dB, 3 dB bandwidth greater than 5.4 GHz, and saturation powers up to -98 dBm. This amplifier design is well suited for multiplexed readout of quantum circuits or astronomical detectors in a compact configuration which can foster on-chip implementations.