RF / S-Parameters

$\Gamma=(Z_L-Z_0)/(Z_L+Z_0)$ becomes zero exactly when $Z_L=Z_0$ — in this case all incident power transfers to the load, no reflected wave is generated. This is precisely the main goal of matching network design.

$\text{VSWR}=(1+|\Gamma|)/(1-|\Gamma|)$ — this is a practical, easily measured quantity (only amplitude measurement is needed, not phase), so it is widely used in RF engineering, even though Γ itself carries more information (including phase).

The Smith chart is a conformal mapping of the complex plane, where lines of constant resistance and reactance become circles/arcs — this lets you visually see how impedance "moves" as frequency changes or as matching elements are added.

The impedances of inductors and capacitors ($j\omega L$ and $1/j\omega C$) change with frequency — so a network giving exact $\Gamma=0$ at one frequency only works approximately at nearby frequencies. This is a natural trade-off in simple (minimal-component-count) matching networks.