Smith Chart
The Smith chart maps the complex impedance plane onto the unit circle of the reflection coefficient Γ. Invented by Phillip Hagar Smith in 1939, it remains the essential graphical tool in RF and microwave engineering. Click anywhere on the chart to set an impedance point, see the corresponding VSWR and return loss, transform it along a transmission line, and build matching networks with lumped L and C components.
Γ = (Z − Z0) / (Z + Z0) · VSWR = (1 + |Γ|) / (1 − |Γ|) · Z0 = 50Ω
The Smith chart
Phillip Hagar Smith of Bell Telephone Laboratories invented this chart in 1939 as a graphical aid for solving transmission line problems. The chart maps the normalized impedance z = Z/Z0 = r + jx to the complex reflection coefficient Γ through the bilinear (Möbius) transformation Γ = (z − 1)/(z + 1). The key insight is that constant-resistance circles and constant-reactance arcs in the impedance plane map to circles in the Γ-plane, all fitting neatly inside the unit circle |Γ| ≤ 1 (for passive impedances).
Reading the chart
The horizontal axis represents purely real impedances: the center is the reference impedance Z0 (typically 50Ω), the left edge is a short circuit (Z = 0), and the right edge is an open circuit (Z = ∞). The upper half represents inductive (positive) reactance, the lower half capacitive (negative) reactance. The constant-r circles are centered on the real axis and all pass through the open-circuit point. The constant-x arcs are portions of circles centered on the vertical line through the open-circuit point.
Transmission line transformation
Moving along a lossless transmission line of electrical length βℓ rotates Γ by an angle −2βℓ (clockwise for moving toward the generator). On the Smith chart, this is simply rotation around the center. One complete revolution (360°) corresponds to a half-wavelength of line. The outer scale of the chart is calibrated in fractions of a wavelength for this purpose.
Impedance matching
The goal of impedance matching is to transform a load impedance to the center of the chart (Z = Z0, or Γ = 0). Adding a series reactive element moves the impedance along a constant-resistance circle. Adding a shunt (parallel) element is best visualized on the admittance chart (the 180°-rotated Smith chart), where it moves along a constant-conductance circle. Common matching topologies include L-networks (one series + one shunt element), which can match any impedance to Z0 using at most two reactive components.