Chart Analysis Enhanced with Mathcad ®

The graphic analysis used in RF circuit design has historically benefitted from the invention of the Smith® Chart [1, 2]. Now, with the advent of math packages such as Matlab®, Mathematica®, Mathcad® and others, further enhancements are possible. This article reviews classic design examples. With the capabilities of packages such as Mathcad, a designer can improve productivity and enhance analysis capability with minimum effort. This article discusses the generation of the chart in Mathcad; drawing on the chart within Mathcad; and solving circuit design problems. In addition, Mathcad shows the capability of comparative analysis (for example, device scattering parameter comparisons), modeling parasitics in devices, extracting Q, device stability, mapping, use of the extended chart and other RF design capabilities. Many of these capabilities benefit directly from the ease in which the math package handles complex numbers, coupled with the ease of presenting output in graphic form. Numerous discussions have occurred within the Mathcad collaboratory and other sites [3, 4] addressing chart construction and techniques. Mathematica is an excellent example of some of the possible RF design techniques [6]. Some of the earlier approaches were too complicated and impeded the real utility of the chart. Either the programming detail was long and inflexible, or the axis titles required in the graphics display mode were so numerous [5] that additional desired data to be displayed was lost in the maze. A return to the fundamental definition and construction of the chart discussed in this article streamlined the process. Chart construction in Mathcad Constructing the chart in Mathcad is a threestep process. It is illustrated in Appendix A, shown on page 56. First, we generate the “real contour circles.” Second, we generate the “single real axis.” Finally, we generate the “reactance contours.” If the inductance and capacitance regions are treated as separate constructs, four labels are required for graphics output. Although the inductance and capacitance contours are mirror images, the separate constructions have been left intact for clarity. Minimizing the number of required labels keeps the display overhead down and provides space for the desired data to be plotted. The chart is constructed in a rectilinear system instead of polar, therefore entry into the chart requires conversion from a polar coordinate system to a rectilinear system. Data entry could include circle locations for power gain, noise figure, stability, reflection coefficient circles that maximize output power and generalized feedback mapping contours. The chart requires manipulation of the complex reflection coefficient G. The location of G will be given as a vector with magnitude and angle. In addition, for general applications, a complete description for the location of G should be provided in the form of a circular location, including a value for center and radius. All of these possibilities are readily handled in Mathcad with the following equations: