General Reflectance

Let $R(s)$ denote a general impedance. Then the wave variable decomposition in (J.7) gives

$$F(s) = R(s) V(s)$$ (J.13)

$$\,\,\implies\,\,F^{+}(s) + F^{-}(s) = R(s) \left[V^{+}(s) + V^{-}(s)\right]$$ (J.14)

$$= R(s) \left[\frac{F^{+}(s) - F^{-}(s)}{R_0}\right]$$ (J.15)

$$\,\,\implies\,\,F^{-}(s) \left[\frac{R(s)}{R_0}+1\right] = F^{+}(s) \left[\frac{R(s)}{R_0}-1\right]$$ (J.16)

$$\,\,\implies\,\,F^{-}(s) = F^{+}(s) \left[\frac{R(s)-R_0}{R(s)+R_0}\right]$$ (J.17)

$$\isdef F^{+}(s) S(s)$$ (J.18)

Formally, $S(s)$ is the reflectance of impedance $R(s)$ relative to $R_0$. For example, if a transmission line with characteristic impedance $R_0$ were terminated in a lumped impedance $R(s)$, the reflection transfer function at the termination would be $S(s)$. The interpretation of $S(s)$ as a reflectance is shown as a wave flow diagram in Fig. J.17c.

Figure J.17: Three different types of diagram for a basic impedance relation: a) Impedance diagram. b) System block diagram. c) Wave flow diagram.

We are working with reflectance for force waves. Using the elementary relations (J.7), i.e., $F^{+}(s) = R_0V^{+}(s)$ and $F^{-}(s) = -R_0V^{-}(s)$, we immediately obtain the corresponding velocity-wave reflectance

$$\frac{V^{-}(s)}{V^{+}(s)} = \frac{-F^{-}(s)/R_0}{F^{+}(s)/R_0} = - \frac{F^{-}(s)}{F^{+}(s)} = - S(s)$$

Thus, velocity reflectance is simply the negative of force reflectance.