A reflection coefficient (Γ) trace on the Smith chart. The bright dot is the current operating point.
The Core Idea
The Smith chart is the RF engineer's most iconic tool — a graphical calculator that maps the entire infinite impedance plane onto a finite circle. Every point inside the unit circle represents a physically realisable impedance; the distance from centre tells you the reflection coefficient magnitude; the angle tells you the phase. It turns complex impedance matching calculations into geometry.
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.
Lower frequency — the point moves toward the capacitive region (lower half).
The constant-r circles
Circle centre: (r/(r+1), 0) Circle radius: 1/(r+1)
The r = 0 circle is the unit circle (pure reactance). The r = 1 circle passes through the origin (matched). As r → ∞, the circle shrinks to the point (1, 0) — an open circuit.
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.
Higher frequency — the point moves into the inductive region (upper half). A series RLC traces a clockwise circle as frequency sweeps upward.
Reading the chart
- Centre (Γ = 0): Perfect match. Z = Z₀. No reflected power.
- Right edge (Γ = 1): Open circuit. All power reflected, in phase.
- Left edge (Γ = −1): Short circuit. All reflected, 180° out of phase.
- Upper half: Inductive (positive reactance).
- Lower half: Capacitive (negative reactance).
Further reading
- Smith chart (Wikipedia)
- Gonzalez, Microwave Transistor Amplifiers — Chapter 2 is the definitive tutorial.
- Pozar, Microwave Engineering