Chapter 3: The Smith Chart
A transmission line changes its impedance depending on material properties and geometric dimensions. Typical practical realizations include microstrip line, coaxial cable, and parallel-plate line. In addition, both the length and operating frequency of the transmission line significantly influence the input impedance. In the previous chapter we derived the fundamental equation describing the input impedance of a terminated transmission line. We found that this equation incorporates the characteristic line impedance, load impedance, and, through the argument of the tangent function, line length and operating frequency. As we saw in Section 2.9, the input impedance can equivalently be evaluated by using the spatially dependent reflection coefficient. To facilitate the evaluation of the reflection coefficient, P. H. Smith developed a graphical procedure based on conformal mapping principles. This approach permits an easy and intuitive display of the reflection coefficient as well as the line impedance in one single graph. Although this graphical procedure, nowadays known as the Smith Chart, was developed in the 1930s prior to the computer age, it has retained its popularity and today can be found in every data book describing passive and active RF/MW components and systems. Almost all computer-aided design programs utilize the Smith Chart for the analysis of circuit impedances, design of matching networks, and computations of noise figures, gain, and stability circles. Even instruments such as the ubiquitous network analyzer have the option to represent certain measurements in a Smith Chart format.
This chapter reviews the steps necessary to convert the input impedance in its standard complex plane into a suitable complex reflection coefficient representation via a specific conformal transformation originally proposed by Smith. The graphical display of the reflection coefficient in this new complex plane can then be utilized directly to find the input impedance of the transmission line. Moreover, the Smith Chart facilitates evaluation of more complicated circuit configurations, which will be employed in subsequent chapters to build filters and matching networks for active devices.
The following sections present a step-by-step derivation of the Smith Chart followed by several examples of how to use this graphical design tool in computing the impedance of passive circuits.