Probelm 9.3
Consider the propagation of a time-harmonic (sinusoidal) wave of frequency $\omega$ in the steady state along a
TEM transmission line characterized by the line parameters {$R', C', L', G'$}. Let the phasor line voltage and
phasor line current be denoted by $\bar{V}(z)$ and $\bar{I}(z)$, respectively.
(a) Derive the telegrapher's equations that govern $\bar{V}(z)$ and $\bar{I}(z)$. Then find the complex wave equation that
is satisfied by both. Express the propagation constant $\gamma = \alpha + j\beta$ by determining the attenuation constant
and phase constant explicitly.
(b) Given $V_0^+$ and $V_0^-$ are complex wave amplitudes, verify that $\bar{V}(z) = V_0^+e^{-\gamma z} + V_0^-e^{+\gamma z}$ satisfies the
complex wave equation. Then interpret the two additive components of $\bar{V}(z)$.
(c) Let $Z_0$ denote the characteristic impedance of the TEM line. Express $\bar{I}(z)$ in terms of $V_0^+$, $V_0^-$, $Z_0$ and
$\gamma$. Determine $Z_0$ explicitly at the wave frequency $\omega$ using the telegrapher's equations.
