A 3-D Smith Chart Based on the Riemann Sphere for Active and Passive Microwave Circuits

This letter proposes a spherical 3-D Smith Chart suitable for representing both active and passive microwave circuits. Using the mathematical concept of the Riemann sphere, the extended reflection coefficient plane is transformed into the surface of the unit sphere. Since the proposed Smith Chart compiles the whole complex plane, all possible loads are included. A simple graphic tool is thus obtained that successfully unifies active and passive circuits. In addition, lossy lines with complex characteristic impedances can also be represented. The letter presents the 3-D Smith Chart, provides its main governing equations, and also enumerates its more important properties.