The Mathematics
The reflection coefficient relates impedance Z to the reference impedance Z₀:
Γ = (Z − Z₀) / (Z + Z₀)
Normalise: z = Z/Z₀ = r + jx. Then:
Γ = (z − 1) / (z + 1) = (r − 1 + jx) / (r + 1 + jx)
|Γ| ≤ 1 for all passive impedances, so every physically realisable impedance maps inside the unit circle. The Smith chart plots this directly: constant-resistance circles and constant-reactance arcs form the coordinate grid.
Impedance matching
The goal: move Γ to the centre (Γ = 0, perfect match). Two fundamental moves:
- Series element: Move along a constant-resistance circle.
- Shunt element: Move along a constant-conductance circle.
Switch the Type control between the impedance and admittance paths to see each move in its natural coordinate grid — series elements read cleanly on the impedance chart, shunt elements on the admittance chart (which is just the impedance chart rotated 180°).
Further reading
- Smith chart (Wikipedia)
- Gonzalez, Microwave Transistor Amplifiers — Chapter 2 is the definitive tutorial.
- Pozar, Microwave Engineering