Low-loss transmission-line attenuation ($\alpha =R^{\prime }/(2Z_0)+G^{\prime }{Z}_0/2$)

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.