Reinterpreting the Smith Chart Using Conformal Geometric Algebra

Abstract

In this study, transmission line calculations using the Smith chart were reinterpreted within the context of Conformal Geometric Algebra (CGA). Reflection coefficients and immittances (representing either impedances or admittances depending on the context) can be straightforwardly represented in CGA, together with standard Smith chart impedance/admittance transformations. Both single- and double-stub tuners (matching networks) were investigated to assess the efficacy of this approach. All geometric calculations inherent in a paper-based calculation using the Smith chart have been replicated in CGA, in which lines (radials) and circles are represented by multivector representations of planes and spheres respectively. Standard CGA operators are then applied to these multivectors to determine intersections, from which angles and lengths (needed in transmission line network specifications) can be calculated. Although appearing to be algebraically complex, the CGA calculations are fundamentally simple in form, and can be readily calculated using an appropriate computer-based toolkit. This observation suggests that CGA can be regarded as a useful augmentation to traditional vector analysis, and can benefit both students and practitioners in synthesizing transmission line networks that would traditionally have been solved using a Smith chart, provided they have a background in geometric algebra fundamentals and a suitable computer-based toolkit is available.

Keywords

• Conformal geometric algebra
• Smith chart
• transmission lines