The Problem
Before the Smith Chart makes sense, you need to feel why it matters. When a signal travels down a transmission line and hits a load with a different impedance, some of the power bounces back. Drag the slider to see how much.
Perfect match — zero reflection. All power reaches the load.
The fraction that bounces back is determined by the reflection coefficient Γ = (Z − Z₀) / (Z + Z₀). At R = Z₀, Γ = 0. At a short circuit (R = 0) or open circuit (R = ∞), |Γ| = 1. Everything in between is what the Smith Chart is for.
Impedance as a Point
Every complex impedance Z = R + jX maps to exactly one point in the complex Γ plane via Γ = (Z − Z₀) / (Z + Z₀). Drag the sliders to move your impedance and watch where Γ lands. The dashed blue circle is the constant-VSWR circle passing through that point.
Positive reactance (inductive) — the point is in the upper half.
Notice: the unit circle |Γ| = 1 is the outer boundary. Points outside it would require negative resistance — physically impossible for a passive load. Every passive impedance maps inside or on the unit circle. That boundary is the edge of the Smith Chart.
The Grid Revealed
The Smith Chart is just the Γ plane with a special coordinate grid overlaid on it — one that makes impedance values easy to read directly. Click below to build it layer by layer.
Start with just the outer boundary — the unit circle |Γ| = 1. Every Smith Chart ever drawn is built on this foundation.
The grid lines are not arbitrary. Every constant-r circle and constant-x arc is derived directly from the Γ = (z−1)/(z+1) mapping. Smith didn't invent a new coordinate system — he just drew the right circles so engineers didn't have to do complex arithmetic by hand.
Reading the Chart
Now drag the impedance point anywhere on the chart. Every position gives you instant R, X, VSWR, and return loss. Try the landmark presets to see where the textbook special cases land.
Landmark presets
The dashed blue circle passing through your point is the VSWR circle — all impedances on it have the same VSWR and return loss. Moving along a lossless transmission line traces exactly this circle. That's what the next section shows.
Transmission Lines
When you add a lossless transmission line between source and load, the impedance seen at the source end is the load impedance rotatedaround the VSWR circle. Scrub the length below — or press Play — to see it happen.
At θ=0° you're looking directly at the load: Z_in = Z_load = 25+30j Ω.
Two lengths matter most: λ/4 rotates exactly 180° — a quarter-wave transformer turns a high impedance into a low one and vice versa. λ/2 completes a full revolution and returns to the original impedance. That's why λ/2 stubs are transparent on the Smith Chart.
Matching Networks
Now put it together. The load below is Z = 25+j30Ω — a real antenna impedance from a poorly matched LNA input. Your goal: design an L-network (one series element + one shunt element) that transforms it to 50Ω. Drag the sliders and watch the path trace across the chart.
No series element
No shunt element
This two-element L-network is the simplest matching topology. For broader bandwidth you'd use a Pi or T network (more elements, more degrees of freedom). But the principle is always the same: navigate the Smith Chart grid with series moves (along r-circles) and shunt moves (along g-circles) until you reach the center.
Ready to calculate with your own impedances?
Use the full Smith Chart calculator — enter any load impedance and reference impedance, get exact Γ, VSWR, return loss, and mismatch loss.
Open Smith Chart Calculator →