Applications of Conformal Geometric Algebra to Transmission Line Theory

Abstract

In this paper, we present the application of a projective geometry tool known as conformal geometric algebra (CGA) to transmission line theory. Explicit relationships between the Smith Chart, Riemann Sphere, and CGA are developed to illustrate the evolution of projective geometry in transmission line theory. By using CGA, fundamental network operations, such as adding impedance, admittance, and changing lines impedance can be implemented with rotations, and are shown to form a group. In addition, the transformations relating different circuit representations, such as impedance, admittance, and reflection coefficient are also related by rotations. Thus, the majority of relationships in transmission line theory are linearized. Conventional transmission line formulas are replaced with an operator-based framework. Use of the framework is demonstrated by analyzing the distributed element model and solving some impedance matching problems.

Keywords

• Transmission lines
• network theory
• network design
• geometry
• applied mathematics