Smith Chart Basics

Upon initial look, the Smith chart appears to be quite complicated. Although the basis for the chart is deeply rooted in complex variable theory, you do not have to be a graduate-level mathematician to appreciate and have a solid working knowledge of the chart. While calculation methods can be performed via computer or calculator with a greater degree of accuracy, the Smith chart still remains a venerable and useful tool for the RF engineer.

One good working definition of the Smith chart is that it is simply a graphical calculator for normalized impedance and associated RF parameters. When impedance is normalized, it is related to a reference value as a multiple. For obvious reasons values of 50 and 75Ω are the most common bases utilized. In the simplest form, if 50Ω is our reference value, then 50, 100 and 25Ω normalized would become 1Ω, 2Ω and 0.5Ω respectively.

Impedance is a measure of the opposition of the flow of current in an alternating current circuit. It is the complex sum of the resistance and reactance components, and thus, in ac circuits describes not only the ratio of voltage and current, but also the relative phase angles. Impedance can be written a number of different ways; however, you probably will see the Cartesian forms below more often than not.

The difference in these two equations results from the flavor of reactance. Inductive reactance is defined as positive, while capacitive reactance is negative. The reason for the sign change is how voltage and current are related in inductors and capacitors. Recall “ELI the ICE man” and that voltage leads current by 90 degrees in an inductor and vice-versa in a capacitor. Finally, note we are only considering positive resistance values. Negative resistance values do occur with some regularity in directional antenna systems; however, they are not manifested in passive circuits and not represented on the basic Smith chart as we will infer shortly.

Figure 1. The Smith chart (Click image to enlarge.)

The details

The Smith chart (Figure 1) is constructed from three families of circles. The first family of circles is tangent to a point on the right side of the outer circle along the center horizontal line. Each of these circles has a constant normalized resistance value. As one moves along any particular circle, the resistance value of the impedance will remain constant, but the reactance will vary.

The second family of circles is that of constant reactance, which are tangent to the same point as the constant resistance circles but are rotated clockwise or counter-clockwise by 90 degrees. These circles form what appear to be arcs on the basic chart with positive or inductive reactance lying above the center line and the converse for capacitive or negative reactance. Reactance is a frequency dependent quantity and can be calculated by the following:

The third set of circles is not normally visible on typical versions of the chart save for one element. These circles form the constant reflection coefficient circles and are concentric radiating out from the center of the chart. Although not strictly correct, these circles can also be thought of as constant VSWR circles because of the relationship between the reflection coefficient and VSWR.

The reflection coefficient varies from a maximum of -1 through 0 to 1. A perfect short is realized by a reflection coefficient of -1, while 1 represents a perfect open, and zero a perfect match. Since a perfect open or short is not usually realized, it is usually more accurate to represent the reflection coefficient, denoted by Γ, with a phase angle.

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