Matrix theory for n-line mxicrowave coupler design

The matrix equations respresenting wave propagation along a multiconductor system of transmission lines are solved giving expressions for the scattering parameters in terms of the parameters of the lines themselves. We demonstrate the extensions and generalisations of co- and contradirectional coupling, coupled-mode theory and also the concept of the non-mode-converting termination to multiline systems. Concentrating on the simplifying assumption of weak reflections at the terminations we extend the weak coupling approximation of coupled-mode theory to allow for strong coupling between adjacent lines. A demonstration of this is given in the design of a three-line equal power splitting coupler with an arbitrary phase shift between output ports at the centre frequency.