Darlington's theorem and complex normalization

Abstract

SUMMARYTwo novel proofs are presented to establish the equivalence between complex normalization and Darlington representation of the loads. Schwarz reflection of analytic functions and Hermitian metrics serve in these proofs as natural concepts for the analysis of lossless multiports. By use of these tools, the algebraic structure that underlies the physical notion of losslessness is analysed, and a class of matrix representations of lossless 2n-ports is derived. The fundamental transformations between these matrices are established to facilitate easy conversion of results related to different choices of co-ordinates (voltages/currents, waves, impedances, reflectances). By virtue of this theory, complex normalization is tacitly extended to include not only impedances but also reflectances as references. Explicit expressions for a 2n-port normalized to complex loads are derived. Invariance of the complex normalized scattering matrix in a lossless cascade circuit is shown to be the most comprehensive generalization of the well known invariance of scalar reflection magnitudes. The relation between this viewpoint and the general theory of invariants of linear fractional maps in terms of cross-ratios is pointed out explicitly. The factorization problem, common to complex normalization as well as Darlington representations, is also reviewed, and it is shown that the concept of minimal factorization (well known in linear prediction theory) leads to more satisfactory results than those of established factorization recipes in complex normalization.

Network modelling of a class of Surface Acoustic Wave filters

“…For a thorough discussion of non-minimal Darlington embeddings in terms of poles and zeros of a scalar impedance, see [29,Section 6]. Most notably, [29, equation (78)] defines a P-matrix embedding of a positive real impedance function instead of an admittance function.…”