Reflectances of Elementary Impedances

Capacitor Reflectance.

For a capacitor of $C$ Farads, the driving-point impedance is (see §J.1.3)

$$R_C(s)=\frac{1}{Cs}$$

(or $k/s$ for a spring with constant $k$). Substituting into Eq. (N.1) gives the reflectance

$$S_C(s) = \frac{R_C(s)-R_0}{R_C(s)+R_0} = \frac{1 - R_0 C s}{1 + R_0 C s} \protect$$ (N.5)

Inductor Reflectance.

For an inductor of $L$ Henrys, we have

$$R_L(s) = Ls$$

$$\,\,\implies\,\,S_L(s) = \frac{Ls-R_0}{Ls+R_0} = \frac{ s - R_0/L }{ s + R_0/L} \protect$$ (N.6)

Resistor Reflectance.

Finally, for a resistor of $R$ Ohms, we get

$$S_R(s) = \frac{R-R_0}{R+R_0} = \frac{1 - R_0/R }{ 1 + R_0/R } \protect$$ (N.7)

Note that both the capacitor and inductor reflectances are stable allpass filters, as they must be. Also, the resistor reflectance is always less than 1, no matter what waveguide impedance $R_0>0$ we choose.