Reflection Γ — Impedance Matching

Part 3 · Impedance Matching

Reflection coefficient \(\Gamma\)

Key point: relative to \(Z_0\), every load is a point on the unit disk. \(1-|\Gamma|^2\) is the fraction of available power the load takes.

\(Z_0 = 50~\Omega\). Center is the match. The dashed ring is this \(|\Gamma|\).

1
Matched.
50 + j0 into 50 Ω. \(\Gamma\) is 0.000 + j0.000. \(|\Gamma|\) is 0.000. Delivered fraction is 100.0%. VSWR is 1.00. The dot is the green center.
2
Short, left rim.
\(\Gamma\) is −1.000 + j0.000. \(|\Gamma|\) is 1.000. Delivered is 0.0%. Return loss is 0.00 dB. VSWR is ∞. Everything bounces.
3
Open, right rim.
\(\Gamma\) is 1.000 + j0.000. Same \(|\Gamma|\) of 1.000, same 0.0% delivered, VSWR ∞. The bounce flipped ends of the chart.
4
100 + j25.
\(\Gamma\) is 0.351 + j0.108. \(|\Gamma|\) is 0.368. Delivered is 86.5%. Return loss is 8.69 dB. \(S_{11}\) is −8.69 dB. VSWR is 2.16. Off the real axis.
5
25 − j10.
\(\Gamma\) is −0.310 − j0.175. \(|\Gamma|\) is 0.356, close to the last point, other quadrant. Delivered is 87.3%. Return loss is 8.97 dB. VSWR is 2.10.

Part 4 is the allowed strokes on this disk. Series rides constant \(R\). Shunt rides constant \(G\).

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—

\(\Gamma\), return loss, and VSWR

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

= —

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

= —

\[ |\Gamma| \]

= —

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

= —

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

= —

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

= —