Reflection Γ — Impedance Matching

Along the way

1
Normalize first. \(z = Z_L / Z_0\). The chart (and \(\Gamma\)) live in this dimensionless plane. \(Z_0\) is usually 50 Ω — change it and the same ohms land somewhere else.
2
\(\Gamma\) maps the right-half \(Z\) plane into the disk. Center = match (\(z = 1\)). Left rim = short. Right rim = open. Passive loads stay inside (or on) the unit circle.
3
\(1 - |\Gamma|^2\) is Part 1’s fraction. For a real \(Z_0\) source, that is \(P_L / P_{\mathrm{avail}}\). Same story, now in bounce language.
4
Return loss and VSWR are just \(|\Gamma|\). \(\mathrm{RL} = -20\log_{10}|\Gamma|\) (positive dB). Analyzers plot \(S_{11}\) as the negative of that. VSWR \(\to 1\) as you approach the center.
5
Reactance rotates off the real axis. Add \(+jX\) or \(-jX\) and the point leaves the horizontal diameter — same \(|\Gamma|\) family (dashed blue circle).

Green center = match. Red → green as \(|\Gamma|\) shrinks. The dashed blue ring through the point is constant \(|\Gamma|\) (constant RL and VSWR).

Next: Part 4 — allowed strokes on this disk (series vs shunt).

\(\Gamma\) from \(Z_L\) and \(Z_0\)

Unit disk · green center = match · dashed ring = this \(|\Gamma|\)

\(Z_0\) (Ω) \(R_L\) (Ω) \(X_L\) (Ω)

Γ plane · unit circle · this \(|\Gamma|\)

Γ (re, im)—

|Γ|—

1 − |Γ|²—

Return loss—

S₁₁—

VSWR—

\[ z = \dfrac{Z_L}{Z_0} \]

= —

\[ \Gamma = \dfrac{z - 1}{z + 1} \]

= —

\[ |\Gamma| \]

= —

\[ 1 - |\Gamma|^2 \]

= —

\[ \mathrm{RL} = -20\log_{10}|\Gamma| \]

= —

\[ \mathrm{VSWR} = \dfrac{1 + |\Gamma|}{1 - |\Gamma|} \]

= —

Try it: Matched → center, \(1-|\Gamma|^2 = 100\%\). Short or open → rim, delivered → 0. Then add \(+jX\) on a matched \(R_L\) — you leave the real axis on a constant-\(|\Gamma|\) circle. Same bounce, new phase.

Analog Intuition

Reflection coefficient \(\Gamma\)

\(\Gamma\) from \(Z_L\) and \(Z_0\)