Beltrami-Klein disk model as viewed for use in impedance trajectory projection

Abstract

This paper presents a basic consideration of complex impedance projection. We start by recalling the traditional Smith chart. The well-known standing wave ratio is revisited by its natural logarithm, which brings Poincaré metric to geometrical distance on the chart. We then replace the distance by its double to observe the Beltrami-Klein disk model from the aspect of RF engineering. To give fundamental locus examples, we formulate how lumped-constant elements behave on hyperbolic coordinates, and find that they draw their constant-R and -X contours in straight lines or vertical ellipses. For higher frequency applications, we also formulate the behavior of transmission lines to show what trajectories they exhibit on the disk.