Physical Derivation of Reflection Coefficient

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

$$\Gamma _J(i) \isdef \sum_{i\neq j} \Gamma _i \protect$$ (F.21)

denote this parallel combination, in admittance form. Then we must have

$$\rho_i = \frac{R_J(i)-R_i}{R_J(i)+R_i} = \frac{\Gamma _i-\Gamma _J(i)}{\Gamma _i+\Gamma _J(i)} \protect$$ (F.22)

Let's check this ``physical'' derivation against the formal definition Eq.(F.17) leading to $\rho_i = \alpha_i - 1$ in Eq.(F.19). Toward this goal, let

$$\Gamma _J \isdef \sum_{j=1}^N \Gamma _j$$

denote the parallel combination of all admittances connected to the junction. Then by Eq.(F.21), we have $\Gamma _J = \Gamma _i + \Gamma _J(i)$ for all $i$ . Now, from Eq.(F.14),

$$\begin{eqnarray*} \rho_i &\isdef & \alpha_i - 1 \;\isdef \; \frac{2\Gamma _i}{\Gamma _J} - 1\\ &\isdef & \frac{2\Gamma _i - \Gamma _J}{\Gamma _J} \;=\; \frac{2\Gamma _i - \left[\Gamma _i + \Gamma _J(i)\right]}{\Gamma _i + \Gamma _J(i)}\\ &=& \frac{\Gamma _i - \Gamma _J(i)}{\Gamma _i + \Gamma _J(i)} \;=\; \frac{R_J(i) - R_i}{R_J(i)-R_i} \end{eqnarray*}$$

and the result is verified.