Reflectance of an Impedance

Let $R(s)$ denote the driving-point impedance of an arbitrary continuous-time LTI system. Then, by definition, $F(s)=R(s)V(s)$ where $F(s)$ and $V(s)$ denote the Laplace transforms of the applied force and resulting velocity, respectively. The wave variable decomposition in Eq.(C.74) gives

$$F(s) = R(s) V(s)$$

$$\,\,\Rightarrow\,\,F^{+}(s) + F^{-}(s) = R(s) \left[V^{+}(s) + V^{-}(s)\right]$$

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

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

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

$$\isdef F^{+}(s)\, \hat{\rho}_f(s) \protect$$ (C.75)

We may call $\hat{\rho}_f(s)$ 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, as seen from the end of the transmission line, would be $\hat{\rho}_f(s)$ .

We are working with reflectance for force waves. Using the elementary relations Eq.(C.73), i.e., $F^{+}(s) = R_0V^{+}(s)$ and $F^{-}(s) = -R_0V^{-}(s)$ , we immediately obtain the corresponding velocity-wave reflectance:

$$\hat{\rho}_v(s) \isdefs \frac{V^{-}(s)}{V^{+}(s)} \eqsp \frac{-F^{-}(s)/R_0}{F^{+}(s)/R_0} \eqsp - \frac{F^{-}(s)}{F^{+}(s)} \eqsp - \hat{\rho}_f(s)$$