Unified Microwave Network Theory Based on Clifford Algebra in Lorentz Space

A unified network theory is presented. It consists of three parts: a network formalization, a geometrical model, which is the Minkowski model of Lorentz space, and a mathematical tool, Clifford algebra. The latter is well suited in dealing with rotations in Lorentz space. The rotations can be represented by the exponentials of a single infinitesimal isometry or a single Clifford bivector. Special emphasis is put on the parabolic rotations. Through the work of M Riesz we now know how to deal with these. The network theory is applied to some simple synthesis examples starting with a given insertion loss power ratio. The entire procedure is performed in Lorentz space. Transformations to Flatland, the impedance plane, for example, are done by simple projective transformations. The examples chosen are: 1) Butterworth-3 network (parabolic), 2) Chebyshev-1 network (hyperbolic-parabolic), 3) simple stepline (hyperbolic-elliptic), and 4) exponentially tapered line (finite continuous transformation group).