Physical Derivation of Series Reflection Coefficient

Physically, the force-wave reflection coefficient seen at port $i$ of a series adaptor is due to an impedance step from $R_i$ , that of the port interface, to a new impedance consisting of the series combination of all other port impedances meeting at the junction. Let

$$R_J(i) \isdef \sum_{i\neq j} R_i \protect$$ (F.42)

denote this series combination. Then we must have, as in Eq.(F.22),

$$\rho_i = \frac{R_J(i)-R_i}{R_J(i)+R_i}$$ (F.43)

Let's check this ``physical'' derivation against the formal definition Eq.(F.38) leading to $\rho^v_i = \beta_i - 1$ in Eq.(F.40). Define the total junction impedance as

$$R_J \isdef \sum_{j=1}^N R_j$$

This is the series combination of all impedances connected to the junction. Then by Eq.(F.42), $R_J = R_i + R_J(i)$ for all $i$ . From Eq.(F.33), the velocity reflection coefficient is given by

$$\begin{eqnarray*} \rho^v_i &\isdef & \beta_i - 1 \;\isdef \; \frac{2R_i}{R_J} - 1\\ &\isdef & \frac{2R_i - R_J}{R_J} \;=\; \frac{2R_i - \left[R_i + R_J(i)\right]}{R_i + R_J(i)}\\ &=& \frac{R_i - R_J(i)}{R_i + R_J(i)}\\ &=& -\rho_i \end{eqnarray*}$$

Since

$$\rho^v_i\isdef \frac{v^{-}_i(n)}{v^{+}_i(n)} = \frac{-f^{{-}}_i(n)/R_i}{f^{{+}}_i(n)/R_i} = - \frac{f^{{-}}_i(n)}{f^{{+}}_i(n)} \isdef -\rho_i$$

the result follows.