Multiport network analysis by matrix renormalisation: extension to four ports

Two- and three-port network analysis by scattering renormalisation, the principle of which was described in earlier papers, is extended to the 4-port case by derivation of the 4-port renormalisation transforms. The transforms are completely general and take into account complex normalising impedances that are different for all ports as well as passive and active nonreciprocal networks. As in previous associated papers, scattering parameter definitions are based on the voltage-wave concept with complex normalisation derived directly from classical transmission line theory in which the characteristic impedance and propagation constant are both complex. A few examples of 4-port circuit analysis are given including a rigorous solution of the 4-port directional coupler. The method employed in deriving the transforms is briefly described.