E9G: The Smith Chart

The Smith chart is a graphical tool for solving transmission line and impedance matching problems without complex mathematics. It maps all possible impedances onto a bounded circular graph, allowing engineers to read off impedance, SWR, stub lengths, and matching network values directly.

This lesson covers the coordinate system of the Smith chart, what resistance circles and reactance arcs represent, how the chart is normalized, constant-SWR circles, wavelength scales, and specific features visible in Figure E9-3.

What the Smith Chart Can Calculate

The Smith chart is used to calculate impedance along transmission lines and SWR values. Common applications include:

• Determining the impedance at any point along a transmission line given the load impedance
• Finding the length and position of an impedance matching stub
• Reading SWR directly from the chart
• Designing matching networks for amplifiers and antennas

The Smith chart does not directly calculate radiation resistance, antenna gain, or propagation characteristics — those require other tools.

Coordinate System

The Smith chart uses a specialized coordinate system made up of two families of curves:

• Resistance circles — circles centered on the horizontal axis, each representing a specific normalized resistance value. All points on a given circle have the same resistance.
• Reactance arcs — arcs that terminate on the outer circle, each representing a specific normalized reactance value. Points on a given arc have the same reactance.

Any impedance Z = R + jX can be plotted as the intersection of the appropriate resistance circle and reactance arc.

Resistance Circles and Reactance Arcs

The two families making up the Smith chart are resistance and reactance:

• The arcs on the Smith chart represent points of constant reactance.
• The circles represent points of constant resistance.

Figure E9-3: Smith Chart Features

Figure E9-3 shows a standard Smith chart. Two exam questions ask about specific features visible in that figure.

Normalization

A Smith chart is normalized to a reference impedance — most commonly 50 Ω. Normalization means reassigning the prime center's impedance value to represent the characteristic impedance of the system. When normalized to 50 Ω, the center of the chart (prime center) represents 50 + j0 Ω — a perfect match. All impedance values plotted are divided by 50 Ω to convert them to normalized form before plotting.

Constant-SWR Circles

During impedance matching network design, a third family of circles — constant-SWR circles — is often added to the Smith chart. These circles are centered on the prime center (perfect match point) and each represents a specific SWR value. An impedance plotted anywhere on a constant-SWR circle produces that SWR when connected to the reference impedance system. This makes it easy to visualize how matching moves the impedance toward the center (SWR = 1).

Wavelength Scales

Along the outer edge of the Smith chart are two wavelength scales (running in opposite directions: "wavelengths toward generator" and "wavelengths toward load"). These scales are calibrated in fractions of transmission line electrical wavelength. Moving around the chart by a given fraction of a wavelength on the scale corresponds to moving the same electrical distance along the transmission line, which allows the chart to be used to find stub lengths and matching point locations directly.

E9G Practice Questions