Recall that nonzero reflection coefficients arise when a propagating wave encounters an impedance mismatch-for example, when a transmission line having a characteristic impedance Z 0 = R 0 + j X 0 is terminated with a load impedance Z L = R L + j X L Þ Z 0. In effect, the Smith chart performs the algebra embodied in equations 2 through 16. The following section shows the mathematical derivation that underlies the Smith chart.
USE THE SMITH CHART TO DETERMINE THE INPUT IMPEDANCE MOVIE
5 provides a Quicktime movie of a rectangular graph of the complex-impedance plane morphing into the polar plot of the typical Smith chart.
To that end, the Smith chart transforms the rectangular grid of the complex impedance plane into a pattern of circles that can directly overlay the complex reflection coefficient plane of Figure 2. Typically, you’ll want to relate reflection coefficients to complex source, line, and load impedances. As noted above, a graph such as Figure 2’s provides for convenient plotting of complex reflection coefficients, but such plots aren’t particularly useful by themselves. It’s the other circles (the gold nonconcentric circles and circle segments in Figure 1) that give the Smith chart its particular value in solving problems and displaying results. The Smith chart resides in the complex plane of reflection coefficient Γ = Γ r + Γ i = | Γ | e j q= | Γ |/_θ.
Figure 2 shows the specific case of a complex G value 0.6 + j0.3 plotted in rectangular as well as polar coordinates (0.67/_26.6°).įigure 2. Note that this latter format omits the absolute-value bars around magnitude G in complex-notation formats that include the angle sign (/_), the preceding variable or constant is assumed to represent magnitude. 4), which is equivalent to the complex reflection coefficient G for single-port microwave components.Īnd that | G |e j q is often expressed as G /_θ. In essence, the Smith chart is a special plot of the complex S-parameter s 11 (Ref. Courtesy of Agilent Technologies.Īlthough the Smith chart can look imposing, it’s nothing more than a special type of 2-D graph, much as polar and semilog and log-log scales constitute special types of 2-D graphs. RF electronic-design-automation programs use Smith charts to display the results of operations such as S-parameter simulation. 3), Smith chart displays can provide an easy-to-decipher picture of the effect of tweaking the settings in a microwave network in an EDA program ( Figure 1), a Smith chart display can graphically show the effect of altering component values.įigure 1. 1, “From the time I could operate a slide rule, I’ve been interested in graphical representations of mathematical relationships.” It’s the insights you can derive from the Smith chart’s graphical representations that keep the chart relevant for today’s instrumentation and design-automation applications. Although calculators and computers can now make short work of the problems the Smith chart was designed to solve, the Smith chart, like other graphical calculation aids (Ref.
1) as a graph-based method of simplifying the complex math (that is, calculations involving variables of the form x + j y ) needed to describe the characteristics of microwave components. Smith (1905–1987) and independently by Mizuhashi Tosaku, is a graphical calculator or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assist in solving problems with transmission lines and matching circuits.The Smith chart appeared in 1939 (Ref. Graphical calculator or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assist in solving problems with transmission lines and matching circuits