Geometric-Analytic Theory of Transition in Electrical Engineering

A geometric-analytic theory of transition is presented and applied to circuit theory. A transition from one state to another is represented in a complex plane by two points which, by variation of a parameter, approach each other, coalesce, and then separate along trajectories perpendicular to the original trajectories. Three analogous cases are treated, namely 1) Movements of fixed points in the complex impedance plane and the complex reflection coefficient plane (Smith chart), 2) Movements of poles in the complex frequency plane, and 3) Movements of saddle points in the complex frequency plane. In the analytic treatment, the linear fractional transformation (Moebius transformation) is used, which makes conformal graphical methods applicable in the geometric treatment. Such a method is, for example, the isometric circle method. By mapping stereographically the complex plane on the Riemann unit sphere, we see that a transition can be represented in three dimensions by the movements of two straight lines, each being the polar of the other with respect to the sphere. The transition takes place when both lines are perpendicular and tangent to the sphere at a point corresponding to the transition point.