Telegrapher's equations ($V_x=-R^{\prime }I-L^{\prime }{I}_t,I_x=-G^{\prime }V-C^{\prime }{V}_t$)

A distributed transmission line with per-unit-length series resistance $R^{\prime }$, series inductance $L^{\prime }$, shunt conductance $G^{\prime }$ and shunt capacitance $C^{\prime }$ obeys the displayed coupled first-order equations by applying Kirchhoff's voltage law and Kirchhoff's current law to a short element. With $R^{\prime }=G^{\prime }=0$, both voltage and current satisfy a wave equation with speed $1/\sqrt{L^{\prime }{C}^{\prime }}$ and characteristic impedance $\sqrt{L^{\prime }/C^{\prime }}$. The quantities $R^{\prime },L^{\prime },G^{\prime },C^{\prime }$ are distributed parameters, not a single lumped resistor or capacitor.

For per-length line parameters $R^{\prime },L^{\prime },G^{\prime },C^{\prime }$ and $e^{-i\omega t}$ convention, a forward factor $e^{-\Gamma z}$ has ${\Gamma }^2=(R^{\prime }-i\omega L^{\prime })(G^{\prime }-i\omega C^{\prime })$. Expanding about the lossless limit with $R^{\prime }\ll \omega L^{\prime }$ and $G^{\prime }\ll \omega C^{\prime }$ gives $\Gamma \simeq \alpha -i\omega \sqrt{L^{\prime }{C}^{\prime }}$, with $Z_0=\sqrt{L^{\prime }/C^{\prime }}$ and the displayed amplitude attenuation. Power decays twice as fast in the exponent. Terminal reflection is a separate power-return mechanism and need not be dissipative.