Transmission Line Impedance Matching

What you are seeing: a coaxial line of characteristic impedance $Z_0 = 50 \, \Omega$ terminated by a load $Z_L$ . The reflection coefficient $\Gamma = (Z_L - Z_0) / (Z_L + Z_0)$ measures the ratio of reflected to incident voltage. At matched load $Z_L = Z_0$ the reflection is zero and all power transfers. At open or short the reflection has magnitude 1 and no power transfers.

Slide $Z_L$ over the resistive range $1 \, \Omega$ to $1000 \, \Omega$ . The two indicators show $|\Gamma|$ and VSWR. The animated standing wave on the line illustrates that mismatched impedances produce a visible voltage envelope (interference between forward and reflected waves); matched loads give a uniform amplitude.

Figure 1. Transmission line voltage standing wave with reflection

\Gamma = (Z_L - Z_0)/(Z_L + Z_0)

. Method: closed-form.

load R_L (Ohm)120

load X_L (Ohm)70

matchingnone

WHAT TO TRY

Drag R_L and X_L away from 50 ohms and watch the load point leave the Smith-chart centre while the live standing wave develops deep nulls; the dashed VSWR circle is the locus the impedance traces as you slide along the line.
Add reactance with X_L: the load point swings up (inductive) or down (capacitive) the constant-resistance arc, and the standing-wave pattern shifts because the reflection picks up a phase.
Set X_L to zero, then switch matching to the quarter-wave transformer: Z_t = sqrt(Z_0 R_L) drives Gamma to zero, the standing wave flattens to a pure travelling wave, and the Smith point snaps to the matched centre with 100 percent power delivered.