Reading the Smith Chart

Part 5 · Impedance Matching

Key point: every point is a \(Z\) and a \(\Gamma\). Gold is this \(R\). Teal is this \(X\). The center is \(Z_0\).

\(Z_0 = 50\,\Omega\). Click inside the disk, or use a preset. The highlighted gold and teal are this point, not the printed grid.

1
Center is the match.
z is 1.000 + j0.000. Z is 50.0 + j0.0. Γ is 0. |Γ| is 0.000, return loss is ∞, and 100.0% is delivered. The red dot sits on the green center.
2
Left rim is a short.
The chart holds the point just inside the disk. z is 0.003 + j0.000. Z is 0.1 + j0.0. Γ is −0.995. |Γ| is 0.995, return loss is 0.0 dB, and 1.0% is delivered.
3
Right rim is an open.
z is 99.000 + j0.000. Z is 4950.0 + j0.0. Γ is 0.980. |Γ| is 0.980, return loss is 0.2 dB, and 4.0% is delivered. Same rim, other side.
4
Twice \(Z_0\), still real.
z is 2.000 + j0.000. Z is 100.0 + j0.0. Γ is 0.333, on the horizontal diameter. |Γ| is 0.333, return loss is 9.5 dB, and 88.9% is delivered. The gold circle is r = 2.
5
Upper half is +X.
z is 1.000 + j1.000. Z is 50.0 + j50.0. Γ is 0.200 + j0.400. |Γ| is 0.447, return loss is 7.0 dB, and 80.0% is delivered. Gold is the r = 1 circle through the center. Teal is this +X.
6
Lower half is −X.
The preset is labeled 0.5 − j0.5. The live point is z 0.724 − j0.690, Z 36.2 − j34.5, Γ 0.000 − j0.400. |Γ| is 0.400, return loss is 8.0 dB, and 84.0% is delivered. Teal is in the capacitive half.

Shunt moves want a Y grid (constant G, constant B). This page draws the Z grid. Part 6’s matcher toggles both. The ESP32 chip pin is one example in that tool.

Click the chart or use a preset

\(Z_0 = 50\,\Omega\) · gold / teal highlight follow the red point

this R (const-R) this X (const-X) match (Z₀)

Smith · click anywhere inside the disk

Normalized z—

Z (Ω)—

Γ—

|Γ| / RL—

1 − |Γ|²—

Highlighted gold circle = this \(R\). Highlighted teal arc = this \(X\). Center = \(Z_0\).

\(z\) from \(\Gamma\)

\[ \Gamma \]

= —

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

= —

\[ Z = z\,Z_0 \]

= —

\[ |\Gamma| \]

= —

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

= —