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I am investigating nonideal directional couplers, in particular the mathematical constraints between the directivity and other S-matrix parameters. If all ports are matched, it's possible to derive, e.g., \$S_{14}(|S_{13}|^2-|S_{24}|^2)=0\$ No matter what the values of \$S_{13}\$ and \$S_{24}\$ are, this will always hold because of unitarity. See, e.g. Pozar, Microwave Engineering 3rd ed., sec. 7.1, eq. 7.9ff.

I am interested in a more general treatment that does not assume that all ports are matched. I'm especially interested in relating the directivity to the input mismatch \$S_{11}\$. Does anyone have any information on these points?

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Self-answering: check out https://doi.org/10.1109/TMTT.2013.2288601. Excellent work!

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