# Bi-SQUID Versus dc SQUID in Flux-Driven Traveling-Wave Parametric Amplifier
## 2 Effect of Finite Size of Artificial Cells  
Usually, the JTWPA designs are based on implementation of the artificial right-handed lines containing Josephson junctions, although the left-handed artificial lines can also be used as recently reported in [17]. The right-handed artificial lines correspond to the existing natural distributed waveguide lines. Therefore, the natural distributed lines are often modeled by the right-handed artificial LC lines under the stipulation that the length of the distributed line pieces described by the lumped LC cells goes to zero. The limit processing (when \(L,C\to 0\)) leads to telegrapher's equations and hence to the differential wave equation [18].  
However, when the lumped cells of the implemented artificial lines model the continuous line pieces of finite length, one should describe the discrete system using discrete-valued equations as it is realized in [19] (versus continuum approximation in [12]) and partly in [16]. Nevertheless, to simplify analysis of the artificial network, many authors consider the structure as a really distributed continuous waveguide line to describe propagation of the pumping, signal and idler waves in the artificial structure using a wave-type equation for a continuous medium. In the approach, wave impedance of the artificial line is considered as a real-value impedance \(Z_{0}\) of the corresponding really distributed waveguide line. However, this is generally incorrect. Figure 1 illustrates simple reasoning which can be used to calculate wave impedance of the long artificial line, consisting of \(Z_{1}\), \(Z_{2}\)\(\Gamma\)-cells. If \(Z_{0}\) is the wave impedance, then it does not change with connection of the same additional cell. This requirement yields in the following equation:  
\[Z_{0}=Z_{1}/2+\sqrt{Z_{1}Z_{2}+Z_{1}^{2}/4} \tag{1}\]  
In case of a conventional dissipativeless cell line with the cell element impedances \(Z_{1}=i\omega L\) and \(Z_{2}=\left(i\omega C\right)^{-1}\), the wave impedance of the artificial line is complex:  
\[Z_{0}=i\omega L/2+\sqrt{L/C-\left(\omega L\right)^{2}/4}=\sqrt{L/C}e^{i\chi} \tag{2}\]  
\[\chi=arcsin\left[\left(\omega L/2\right)/\sqrt{L/C}\right]=arcsin\left[\omega /\left(2/\sqrt{LC}\right)\right] \tag{3}\]  
where the angle \(\chi\) changes from \(\chi=0\) at \(\omega\to 0\) to \(\chi=\pi/2\) at cutoff frequency  
\[\omega_{cut}=2/\sqrt{LC} \tag{4}\]  
Thus, the wave impedance changes from the real value \(\rho=\sqrt{L/C}\) at a low frequency to the imaginary value \(i\rho\) at \(\omega\rightarrow\omega_{cut}\). This fact means that the artificial cell line cannot be matched with any constant real-valued impedance over the total frequency band up to the cutoff frequency. For example, when the cell line is connected to the real-valued impedance \(\rho=\sqrt{L/C}\), the reflection factor is imaginary and frequency-dependent:  
\[\Gamma=\left(\rho-Z_{0}\right)/\left(\rho+Z_{0}\right)=-i\cdot tg(\chi/2) \tag{5}\]  
The absolute magnitude of the factor changes from \(|\Gamma|=0\) at \(\omega\to 0\) to \(|\Gamma|=1\) at \(\omega\rightarrow\omega_{cut}\). Of course, the problem vanishes with decreasing the cells' size (at \(L\to 0\) and \(C\to 0\)), i.e., with limit processing of the distributed waveguide line when \(Z_{0}\rightarrow\rho=\sqrt{L/C}\).  
To mitigate the reflection problem, one could increase the cutoff frequency of the artificial line as compared with frequencies of the pumping, signal and idler signals; however, it additionally leads to the intense depletion of the pump wave due to the unwanted leak of the pump power to the higher harmonics and intermodulation components occurring in the frequency band ranging up to the cutoff frequency. Moreover, the described dependence of the wave impedance on frequency automatically leads to the frequency-dependent phase velocity decreasing with frequency.