Scattering Matrix of 2N-port hybrid directional couplers

The derivation of the scattering matrix of hybrid directional couplers with more than four ports is rather difficult to find in the literature. Some particular cases can be found, but a general form is not yet discussed. The aim of this contribution is to develop a simple procedure to write the 2NX2N S-matrix for hybrid directional couplers with N input and N output ports. This procedure is based on the separation of the phase of the scattering coefficients in two terms. The first is related to the presence of transmission lines, or phase shifters, connected to the coupler ports and the second to the intrinsic nature of the coupler that impose particular phase relationships to the scattering coefficients to ensure that the S-matrix is unitary. These relationships are due to the presence of one polyphase systems of order N or to m polyphase subsystems of order N/m, if N is multiple of m. Finally, it will be shown that 2N port hybrid directional couplers with phase shift equal to 0 or pi are possible only if N is an integer power of 2.