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Digital Edition
The Smith Chart
By<br>H. Ward Silver
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This strange-looking chart turns the complex mathematics of transmission lines into circles and arcs that unlock the mysteries of SWR, stubs, matching networks, and a whole lot more.
The ways and means of transmission lines can be mysterious as you may remember from the January 2016 Ham’s Wireless Workbench. The key to unraveling such mysteries is often having the information presented in a new way — perhaps an analogy or a picture that explains what’s happening “inside the line” in a new and different manner. For many hams, the magic window is a novel and useful type of graph called a Smith chart.
Smith Chart Background
First described by Phillip Smith in 1939, there are a number of QST articles and a detailed Wikipedia page (en.wikipedia.org/wiki/Smith_chart ) on the Smith Chart. They provide as much background as you care to absorb, and deeper discussions than this article can provide. You can read these articles first or use them as references throughout this article. The ARRL has made the QST articles available for downloading at www.arrl.org/smith-chart .
Where Does the Smith Chart Come From?
Before discussing the chart, let’s back up a step. All impedances consist of resistance and reactance. Graphically, these components are represented as a pair of axes at right angles, as in Figure 1 .
FIGURE 1. This graph shows the rectangular coordinates for any impedance. The X axis represents resistance (R) and the Y axis represents reactance (X). Inductive reactance is positive (+Y axis) and capacitive reactance is negative (-Y axis). Positive resistance which represents loss is on the positive X axis.
The horizontal axis represents resistance (R): positive to the right of the origin and negative to the left. The vertical axis represents reactance (X): positive (inductive) above the origin and negative (capacitive) below. R and X are the rectangular coordinates of the impedance. Hold that thought.
In a transmission line, when a wave of RF voltage and current encounters an impedance different from the characteristic impedance of the transmission line, Z0, some of the energy in the wave is reflected back towards the wave’s source. The phase of the voltage and currents making up the reflected wave will differ from those in the incoming or incident wave depending on the value of the impedance causing the reflection.
The incident and reflected voltage and current waves combine at every point along the line. At each point, the combination results in voltage and current with a phase relationship different from either the incident or reflected waves. It is as if the same energy in the line had been applied to an impedance with values of resistance and reactance that create the same phase relationship. If you cut the line at that point and replace the section beyond the cut with actual components of the equivalent impedance, there would be no change to the waves in the remaining section of the line!
The voltages and currents of both waves also vary with distance along the line because of the AC nature of the waves. This results in different combinations of incident and reflected voltages and currents and their equivalent impedance. For example, if the equivalent impedance looks like 5Ω of resistance and +20Ω of reactance at one point, a bit farther along the line the equivalent impedance might be 20Ω of resistance and -5Ω of reactance. That means the point on the graph of resistance and reactance also moves around with position in the line, returning to its original position every half-wavelength (1/2λ) as it turns out.
Smith Chart Construction
The equation describing how that impedance point moves around on a graph of rectangular coordinates is pretty intimidating:
Z = ZO [ (ZL + j ZO tan (βl)) / (ZO + j ZL tan (βl)) ]
and the path it describes on a rectangular graph does not lend itself to easy use. (βl gives the electrical position along the line.) What Smith discovered, however, was that if you distort the rectangular graph in a certain way (called a mapping), the path...