EM-27 · Transmission-line toolkitTool equip

The Smith Chart

The Smith chart is a tool that spreads complex impedance over the disc of the reflection coefficient to read at a glance. Mapping a normalized impedance z = r + jx by Γ = (z−1)/(z+1), every impedance becomes a single point inside radius 1. The very center is the matched point, and moving along the line makes the point circle that center.

Place the impedance as a point on the chart

Adjust the resistance r and the reactance x and the normalized impedance z = r + jx is plotted as a point (gold) on the Smith chart. The farther from the center, the larger the reflection; the closer to the center, the smaller. Set r to 1 and x to 0 to land the point on the matched center (green). See what the reflection coefficient becomes there.

Resistance rr = 2.0

Reactance xx = 2.0

Drag to orbit. The sliders set r and x.

The impedance and reflection on the chart

Γ = (z − 1)/(z + 1) · maps impedance to the reflection coefficient

The center (z = 1) is matched, the right edge open, the left edge short

Impedance z 2.0 + j2.0 · Reflection |Γ| 0.62

Edge · large reflection

Reflection as a picture

In the last unit 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. For a passive load |Γ| ≤ 1, so all those points fall inside a disc of radius 1. The Smith chart is exactly this Γ-plane, overlaid with a curved grid from which the impedance can be read. Devised by Phillip Smith in 1939, it has been an essential tool of high-frequency engineers ever since.

Normalized impedance and Γ

To use one chart for every line, we use the normalized impedance z = Z/Z₀ = r + jx, the impedance divided by the characteristic impedance; r is resistance, x reactance. Mapping this z by Γ = (z−1)/(z+1) gives a point on the chart. The center (z = 1) is Γ = 0, the perfect match. The right edge (z = ∞) is Γ = +1, an open; the left edge (z = 0) is Γ = −1, a short. The upper half is inductive (x > 0), the lower half capacitive (x < 0).

ObserveΓ = (z − 1)(z + 1)

Impedance maps to the reflection coefficient.

Choosez = ZZ₀ = ?

Normalizing divides the impedance by Z₀.

Resistance circles and reactance arcs

The curved grid on the chart lets you read the impedance directly. Points of constant resistance (r = const) form circles, all meeting at the single point on the right edge. Points of constant reactance (x = const) form arcs springing from that same point. Seeing which resistance circle and which reactance arc a point lies on lets you read off its r and x at once. It is a grid for reading impedance by eye, without calculation.

Fill inz = 1 ⇒ Γ = ?

The center of the chart is the matched point.

The point circling along the line

The real power of the Smith chart shows when you move along the line. Going back up the line from the load toward the source, the impedance seen at that point appears as a point rotating clockwise around the center of the chart. Since |Γ| is unchanged, it follows a circle of the same radius. Travel half a wavelength (λ/2) and the point goes all the way around, back to where it started. So where on the line the impedance takes any given value, and where to insert a matching element, can be found right on the chart by rotating around.

On your ownSWR = ?

The reflection coefficient sets the SWR.

Back to the first screen

On the first screen, as you moved the resistance r and the reactance x, the impedance was plotted clearly as a point on the chart, and the farther from the center, the larger the reflection. The moment you set r to 1 and x to 0 to bring the point to the very center, the reflection coefficient became zero. That is the matched point. The Smith chart turns the complex calculation Γ = (z−1)/(z+1) into geometry on a single disc, letting you read impedance, rotate along the line, and design a match — all by eye.

The Smith chart is a tool that maps a normalized impedance z = r + jx by the reflection coefficient Γ = (z−1)/(z+1) to a point on a disc of radius 1. The very center is the matched point (z = 1, Γ = 0); the right edge is an open, the left edge a short. With a grid of resistance circles and reactance arcs you read impedance by eye, and moving along the line makes the point circle the center on a constant-|Γ| circle (one turn per λ/2), so matching can be designed at a glance.

Tool equipOnce you equip this tool

Once the Smith chart is equipped, you no longer compute impedance and reflection separately as complex numbers but handle them on a single picture. The impedance matching of the next unit (EM-28) is, in the end, moving the load’s point to the center on this chart. Seeing how inserting a λ/4 transformer or a stub moves the point on the chart, you find that designing a match is just finding a path on the chart.