Analytical studies supplementing the Smith chart

The study of transmission lines is basic in undergraduate electrical engineering education. In relevant courses, a key subject is the Smith chart (SC), which is a graphical device providing insight. This paper develops certain formulas that can further enhance students' understanding, including an unconventional set of conductance/susceptance transformation equations, as well as simplified versions useful for highly mismatched lines, where it is sometimes difficult to use the SC. All formulas are developed from well-known facts of transmission-line theory. SC interpretations of many of the analytical results are provided. The analytical investigations thus supplement the SC. For highly mismatched lines, the formulas bring out certain resemblances to the resonant series RLC circuit, thus providing an interesting connection to the familiar subject of ordinary circuit theory. Lastly, students can also benefit from the mathematical tools used herein; these include simple ideas of asymptotic analysis.