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 2N×2N 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 coecients 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 imposes particular phase relationships to the scattering coecients 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 π are possible only if N is an integer power of 2.
2. Shelton, J. and K. Kelleher, "Multiple beams from linear arrays," IEEE Trans. Antennas Propag., Vol. 9, No. 2, 154-161, Mar. 1961.
3. Moody, H., "The systematic design of the butler matrix," IEEE Trans. Antennas Propag., Vol. 12, No. 6, 786-788, Nov. 1964.
4. Shelton, J. P., "Fast fourier transforms and butler matrices," Proc. IEEE, Vol. 56, No. 3, 350, Mar. 1968.
5. Ueno, M., "A systematic design formulation for butler matrix applied FFT algorithm," IEEE Trans. Antennas Propag., Vol. 29, No. 3, 496-501, May 1981.
6. Schwelb, O., "Generalized analysis for a class of linear interferometric networks. I. Analysis," IEEE Trans. Microw. Theory Tech., Vol. 46, No. 10, 1399-1408, Oct. 1998.
7. Schwelb, O., "Generalized analysis for a class of linear interferometric networks. Part II: Simulations," IEEE Trans. Microw. Theory Tech., Vol. 46, No. 10, 1409-1418, Oct. 1998.
8. Alessandri, F., M. Giordano, M. Guglielmi, G. Martirano, and F. Vitulli, "A new multiple-tuned six-port riblet-type directional coupler in rectangular waveguide," IEEE Trans. Microw. Theory Tech., Vol. 51, No. 5, 1441-1448, May 2003.
9. Zappelli, L., "Optimization procedure of four-port and six-port directional couplers based on polygon equivalent circuit," IEEE Trans. Microw. Theory Tech., Vol. 66, No. 10, 4471-4481, Oct. 2018.
10. Leclerc, C., H. Aubert, A. Ali, A. Annabi, and M. Romier, "The close-form solution for symmetric butler matrices," Progress In Electromagnetics Research C, Vol. 26, 167-179, 2012.