Analog Intuition

Part 3 · Impedance Matching

Reflection coefficient Γ

One idea: mismatch is a complex number Γ on the unit disk — its size is how much “bounces.”

Γ from ZL and Z0

Unit circle · green center = match · red point = this load

Γ (re, im) —
|Γ| —
Return loss —
VSWR —
\[ z = \dfrac{Z_L}{Z_0} \]
= —
\[ \Gamma = \dfrac{z - 1}{z + 1} \]
= —
\[ |\Gamma| \]
= —
\[ \mathrm{RL} = -20\log_{10}|\Gamma| \]
= —
\[ \mathrm{VSWR} = \dfrac{1 + |\Gamma|}{1 - |\Gamma|} \]
= —

Try it: Matched → Γ at center, RL → ∞. Short/open → rim (|Γ|=1). Add +jX or −jX and watch the point leave the real axis — same |Γ| family as a circle about the origin.

The one idea

Relative to a reference \(Z_0\), every load maps to a point \(\Gamma\) inside (or on) the unit circle. \(|\Gamma| = 0\) means no reflected wave (match). \(|\Gamma| = 1\) means total reflection (open or short on a lossless line).

\[\Gamma = \frac{z - 1}{z + 1},\qquad z = \frac{Z_L}{Z_0}\]

Return loss vs \(S_{11}\) (dB)

Engineers often quote return loss as a positive number:

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

Network analyzers plot \(S_{11}\) in dB as a negative curve (0 at the top). Same information, opposite sign convention — as on the Smith matcher’s \(S_{11}\) chart.

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