A Contribution to the Design of Multi-Element Directional Couplers

A multi-element directional coupler is considered as a cascaded set of 2-element directional couplers, each of which, at the design frequency, is perfect in match and directivity. A characteristic impedance matrix and a voltage transfer matrix may be defined for each of the basic 2-element couplers, and these are analogous to the characteristic impedance and the propagation coefficient in iterative 4-terminal network theory. The voltage transfer matrix is shown to be a linear function of the cosine and sine of a certain angle which is related in a simple way to the coupling factor of the elemental coupler. This function has algebraic properties analogous to those of complex numbers. In particular, the use of de Moivre´s theorem is shown greatly to simplify the design of multi-element couplers. An example is given of the design of a 5-slot 3 dB coupler (a binomial slot hybrid).