It turns out only one multiply is needed to implement three-port adaptors (scattering junctions) having a reflection-free port. The derivation is very similar to deriving the one-multiply two-port scattering junction from the four-multiply Kelly-Lochbaum scattering junction, as discussed in Fig..
Let's begin with the scattering relations in terms of the alpha
parameters introduced in §F.2.2 for a parallel junction
of any number of ports:
and is the wave admittance on port .
Here we wish to focus on the three-port case . To connect with our previous example, set , , and . When port A is reflection-free, we have the constraint that ( ). Scattering relations are unchanged when all port impedances are divided by the same positive number, so divide through by to obtain , and . With , we have , and . In other words, there is only one degree of freedom, for the three-port parallel junction with a reflection-free port.
The alpha parameters become
, so that the scattering formulas simplify to