The Smith Chart

Place the impedance as a point on the chart

Change the resistance r and the reactance x with the two sliders, and the normalized impedance z = r + jx is plotted as a gold point on the Smith chart, joined to the center by a gold line. The longer that line, the larger the reflection; the shorter, the smaller. Set r to 1 and x to 0 and the point lands on the green matched point at the center; what the reflection coefficient becomes then is the question of this lesson.

Reflection as a picture

In the previous lesson the reflection coefficient Γ came out as a complex number for each load. A complex number is a point on a plane, so the Γ of every load can be plotted on one plane, and for passive loads |Γ| ≤ 1, so all those points fall inside a disc of radius 1. The Smith chart is this very Γ plane with curved scales laid over it from which impedances can be read. After Phillip Smith published it in 1939 it became a basic tool for high-frequency engineers, and a little earlier Tosaku Mizuhashi in Japan had put out a chart built on the same idea. Even now that calculators are everywhere, the chart is still used as is on the screens of measuring instruments such as network analyzers.

Normalized impedance and Γ

To use one chart for every line, impedances are divided by the characteristic impedance to give the normalized impedance z = Z/Z₀ = r + jx, where r is the resistance and x the reactance. Mapping z through Γ = (z−1)/(z+1) makes it a point on the chart, and the center (z = 1) has Γ = 0: the perfectly matched point. The right edge (z = ∞) has Γ = +1, an open circuit, and the left edge (z = 0) has Γ = −1, a short circuit. The upper half is inductive (x > 0) and the lower half capacitive (x < 0).

Resistance circles and reactance arcs

The curved scales on the chart let you read impedance directly. Points of constant resistance (r = constant) form circles that all meet at one point on the right edge, and points of constant reactance (x = constant) form arcs fanning out from that endpoint. See which resistance circle and which reactance arc cross at a point, and you can read off that impedance's r and x right away. The outer rim is the circle with no resistance at all, where every load of pure reactance lies.

The point circling along the line

The real power of the Smith chart shows when you follow a line. Moving back along the line from the load toward the source, the impedance seen at each spot appears as a point turning clockwise around the center, and on a lossless line |Γ| stays the same, so it turns along a circle of fixed radius. After half a wavelength (λ/2) the point has gone all the way around and is back where it started. The resistance where this circle crosses the right-hand horizontal axis equals the standing wave ratio, and moving the point to the opposite side of the center gives the reciprocal of the impedance, the admittance, which is what you use for elements added in parallel. So where on the line the impedance takes which value, and where a matching element must go, can be found right on the chart by turning the point.

Back to the first screen

On the first screen, as you moved the resistance r and the reactance x, the impedance was plotted as a point on the chart, and the farther it was from the center, the larger the reflection. The moment you set r to 1 and x to 0 and put the point at the center, the reflection coefficient became 0: that spot is the matched point. The Smith chart turns the complex arithmetic Γ = (z−1)/(z+1) into geometry on a single disc, so reading impedances, turning them along a line, and designing a match can all be done by eye.

With the Smith chart in hand, impedance and reflection are no longer worked out separately as complex numbers but handled on a single picture. The impedance matching of the next lesson (EM-28) also comes down to moving the load's point to the center on this chart. Follow how the point moves on the chart when a λ/4 transformer or a stub is added, and matching design turns out to be a matter of finding a path on the chart.