Normalizing by the Wave Impedance at a Reflection-Free Port

Often we only care about the signal scattering and not the specific impedance values. In that case it is convenient to divide all impedances at the junction by $R_1$ , defining $\tilde{R}_i\isdeftext R_i/R_1$ , so that

$$\begin{eqnarray*} \tilde{R}_1 &=& 1\\ \tilde{R}_i &\in& [0,1], \quad i=2,\ldots,N\\ \sum_{i=2}^{N} \tilde{R}_i &=& 1\\ [5pt] \alpha_1 &=& 1\\ \alpha_i &=& \tilde{R}_i\\ \sum_{i=2}^{N} \alpha_i &=& 1 \end{eqnarray*}$$

String 1 is said to intersect with the other strings at a reflection-free port. For $N>2$ , the other strings are attached to the junction at reflecting ports.