Impedance Matching Network

Designs a two-element L-network to match a complex source impedance Z_S to a complex load impedance Z_L at a single frequency. Up to four solutions (shunt-then-series and series-then-shunt, each with two sign choices) are computed. Matching trajectories are shown on the Smith Chart.

Equations & Parameters ▸

Topology A — Shunt across Z_L, then series toward Z_S
\(B = -B_L \pm \sqrt{G_L\!\left(\tfrac{1}{R_S}-G_L\right)}, \quad X_s = -X_S - \tfrac{B_{\!L,\text{new}}}{G_L^2+B_{\!L,\text{new}}^2}\)
Condition: \(R_S \le |Z_L|^2/R_L\)

Topology B — Series into Z_L, then shunt toward Z_S
\(X = -X_L \pm \sqrt{R_L\!\left(\tfrac{|Z_S|^2}{R_S}-R_L\right)}, \quad B = \tfrac{X_S}{|Z_S|^2}+\tfrac{X_L+X}{R_L^2+(X_L+X)^2}\)
Condition: \(|Z_S|^2/R_S \ge R_L\)

Z_S	Source impedance R_S + jX_S (Ω). The network is designed to present Z_S* to the source.
Z_L	Load impedance R_L + jX_L (Ω). The network is driven into the load.
f	Matching frequency. The L-network provides a conjugate match only at this frequency.
Z₀	Reference impedance for Smith Chart display (does not affect component values).
Q	Network Q ≈ √(R_high/R_low−1). Higher Q means narrower bandwidth.

Source Impedance Z_S

Resistance, R_S

Reactance, X_S

+ inductive, − capacitive

Load Impedance Z_L

Resistance, R_L

Reactance, X_L

+ inductive, − capacitive

Frequency & Reference

Frequency, f

Reference impedance, Z₀

For Smith Chart display

Results

Smith Chart — Matching Trajectories

Reading the diagram
● Square = ZS* (target)
● Triangle = ZL (start)
● Dashed arc = shunt element moves along a constant-conductance circle
● Solid arc = series element moves along a constant-resistance circle
Each solution colour-coded. Moving toward the chart centre = better match.

📖 Learn the theory behind impedance matching →