Steady State Waves

Reflection Coefficient

$\large \Gamma_L$ in a transmission line is given by,

\begin{matrix} Γ_{L} = \frac{Z_{L} - Z_{0}}{Z_{L} + Z_{0}} = ρ e^{j ψ} \\ ρ = \sqrt{\frac{(R_{L} - Z_{0})^{2} + X_{L}^{2}}{(R_{L} + Z_{0})^{2} + X_{L}^{2}}} \\ ψ = a r c t a n (\frac{X_{L}}{R_{L} - Z_{0}}) - a r c t a n (\frac{X_{L}}{R_{L} + Z_{0}}) \end{matrix}

$\large Z_0$ $\large Z_L = R_L + jX_L$ is the load impedance.

$\large Z(z)$ at any point along a transmission line is given by,

\begin{matrix} Z (z) = Z_{0} \frac{Z_{L} - j Z_{0} t a n (β z)}{Z_{0} - j Z_{L} t a n (β z)} \\ β = \frac{2 π}{λ} \end{matrix}

$\large z$ $\large \lambda$ is the wavelength.

$\large S$ is given by,

S = \frac{1 + ρ}{1 - ρ}