HowHowHowHow doesdoesdoesdoes a a a a SmithSmithSmithSmith chartchartchartchart work?work?work?work? Avenerable calculation aid retains its allure in a world of lightning-fast computers and graphical user interfaces. ByByByBy RickRickRickRick Nelson,Nelson,Nelson,Nelson, SeniorSeniorSeniorSenior TechnicalTechnicalTechnicalTechnical EditorEditorEditorEditor The Smith chart appeared in 1939 (Ref. 1) as a graph-based method of simplifying the complex math (that is, calculations involving variables of the formx+ jy)needed to describe the characteristics of microwave components. 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. 2), remains a valuable tool. Smith chart inventor Philip H. Smith explained in Ref. 1, “From the time I could operate a slide rule, Ive been interested in graphical representations of mathematical relationships.” Its the insights you can derive from the Smith charts graphical representations that keep the chart relevant for todays instrumentation and design-automation applications. On instruments (Ref. 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(FigureFigureFigureFigure 1 1 1 1), a Smith chart display can graphically show the effect of altering component values. FigureFigureFigureFigure 1.1. 1.1. RF electronic-design-automation programs use Smith charts to display the results of operations such as S-parameter simulation. Courtesy of Agilent Technologies. Although the Smith chart can look imposing, its 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. In essence, the Smith chart is a special plot of the complex S-parameters11(Ref. 4), which is equivalent to the complex reflection coefficientGfor single-port microwave components. Note that in general, and that|G|ejis often expressed asG/u. Note that this latter format omits the absolute-value bars around magnitudeG; in complex-notation formats that include the angle sign (/), the preceding variable or constant is assumed to represent magnitude. FigureFigureFigureFigure 2 2 2 2 shows the specific case of a complexGvalue 0.6 + j0.3 plotted in rectangular as well as polar coordinates (0.67/26.6). FigureFigureFigureFigure 2.2.2.2.The Smith chart resides in the complex plane of reflection coefficientG=Gr+ Gi= |G|ej= |G|/u.AtpointA,G= 0.6 + j0.3 = 0.67/26.6. WhyWhyWhyWhy thethethethe circles?circles?circles?circles? Thats all well and good, you may say, but where do the Smith charts familiar circles (shown in gold in Figure 1) come from? The outer circle (corresponding to the dashed circle in Figure 2) is easyit corresponds to a reflection coefficient of magnitude 1. Because reflection-coefficient magnitudes must be 1 or less (you cant get more reflected energy than the incident energy you apply), regions outside this circle have no significance for the physical systems the Smith chart is designed to represent. Its 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. Asnoted above, a graph suchasFigure 2sprovides for convenient plotting of complex reflection coefficients, but such plots arent particularly useful by themselves. Typically, youll want to relate reflection coefficients to complex source, line, and load impedances.Tothat end, the Smith chart transforms the rectangular grid of the compleximpedanceplane into a pattern of circles that can directly overlay the complexreflection coefficientplane of Figure 2. Ref. 5 provides a Quicktime movie of a rectangular graph of the complex-impedance plane morphing into the polar plot of the typical Smith chart. The following section shows the mathematical derivation that underlies the Smith chart. In effect, the Smith chart performs the algebra embodied in equations 2 through 16. TheTheTheThe algebraalgebraalgebraalgebra Recall that nonzeroreflection coefficients arise when a propagating wave encountersan FigureFigureFigureFigure 3.3. 3.3. Points of constant resistance form circles on the complex reflection-coefficient plane. Shown here are the circles for various values of load resistance. impedancemismatchforexample,whenatransmissionlinehavingacharacteristic impedanceZ0= R0+ jX0is terminated with a load impedanceZL=RL+ jXLZ0. In that case, the reflection coefficient is In Smith charts, load impedance is often expressed in the dimensionless normalized formzL=rL +xL=ZL/Z0, so Equation 2 becomes Equation 3 is amenable to additional manipulation to obtainzLin terms ofG: Explicitly stating the real and imaginary parts of the complex variables in Equation 4 yields this equation: which can be rearranged to clearly illustrate its real and imaginary components. The f