# Bi-SQUID Versus dc SQUID in Flux-Driven Traveling-Wave Parametric Amplifier
## 4 Bi-SQUID Cell Parameters  
The observed increase in dynamic range when using bi-SQUID cells depends first of all on the magnitudes of the main parameters, which are the normalized cell inductance \(l\) and the normalized critical current of the third Josephson junction \(i_{c3}=I_{c3}/I_{c}\). Both the parameters influence the current redirection inside the cell. Increase in the inductive parameter \(l\), on the one hand, improves the redirection, but on the other hand, a high value of \(l\) decreases the fractional Josephson junction contribution to the cell inductance and hence essentially restricts the obtainable modulation of the reactive parameter. Therefore, the magnitude \(l\sim 1\) can be considered as a tradeoff value of this parameter. Next, Figure 8 shows the 1 dB compression point of the gain of the flux-driven JTWPA with bi-SQUID cells versus the normalized value of the third Josephson junction \(i_{c3}\). An increase in the critical current of this junction results in the monotonic rise of the point, i.e.,  
Figure 6: The current wave amplitude \(I_{p}/I_{c}\) corresponding to the 1 dB compression point of the gain of JTWPA vs. normalized value of the applied dc bias flux \(\varphi_{dc}=2\pi\Phi_{dc}/\Phi_{0}\) with dc SQUID cells (red line) and bi-SQUID cells (black line) at the same value \(l=1\) of the normalized cell inductances. Figure 7: The current wave amplitude \(I_{p}/I_{c}\) corresponding to the 1 dB compression point of the gain of JTWPA vs. normalized value of the applied dc bias flux \(\varphi_{dc}=2\pi\Phi_{dc}/\Phi_{0}\) with dc SQUID cells (red line) and bi-SQUID cells (black line) at the same value \(l=0.75\) of the normalized cell inductances. improvement of the linearity. However, most of the improvement can be achieved with \(i_{\zeta 3}\), \(\sim\)2 to 3. Except the discussed main parameters, one has to consider satellite parameters which can also influence the linear domain of the bi-SQUID cell and hence the gain reduction. These are the second loop inductance and the magnetic flux which can be applied to the loop. Figure 9 schematically shows all inductances in the bi-SQUID cell, where \(L\) is the main part of the principal loop inductance of the cell, and \(L_{dc}\) and \(L_{m}\) are parts of the second loop inductance with the normalized values:  
\[l_{dc}=2\pi l_{c}L_{dc}/\Phi_{0} \tag{13}\]  
\[l_{m}=2\pi l_{c}L_{m}/\Phi_{0} \tag{14}\]  
Unfortunately, inductance of the second loop cannot be decreased exactly down to zero, and some flux can occur in the loop as well. In a rough way, the latter can be considered as the flux \(\Phi_{2}\) of a locally homogeneous magnetic field applied to the bi-SQUID in order to apply to its principal loop dc magnetic flux \(\Phi_{dc}\) needed to set the operation point. In this case, the ratio of the fluxes, \(\kappa=\Phi_{2}/\Phi_{dc}\), can be estimated by the ratio of the loop areas and hence roughly by the ratio of the loop geometric inductances in a square (believing roughly that the loop inductance is proportional to linear size \(r\), while the loop area S\(\sim\)\(r^{2}\)). Interestingly, the moderate normalized inductance component \(l_{dc}\) even slightly rises the 1 dB gain compression points (up to \(\lesssim\) 10%) as shown in Figure 10, but the applied dc flux has an inverse influence and causes a reduction in the overpowering positive impact of the inductance component \(l_{dc}\) as seen in Figure 10.  
As for the other inductance part \(l_{m}\), its increase reduces the 1 dB gain compression points (up to \(\lesssim\) 10%) as seen in Figure 11. These effects can be explained as follows. Inductance \(L_{m}\) is connected in series with the third junction J3 and therefore increases inductive impedance of the branch as compared to the parallel branch containing junctions J1 and J2. On the contrary, the second inductance part \(L_{dc}\) changes the inductive impedance ratio in the reverse direction by increasing the inductive impedance of the J1-J2 branch.