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#### One-Multiply Parallel Reflection-Free Three-Port Adaptor

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:

 (F.23) (F.24)

where denotes the junction force (or voltage), the parameters are defined for parallel junctions by

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 , , and , so that the scattering formulas simplify to

 (F.25) (F.26)

Thus, only one multiply is necessary to compute the junction force (or voltage). In this form we have six additions, but this can be brought down to four. Expand in the last equation to get
 (F.27) (F.28) (F.29) (F.30) (F.31) (F.32)

Thus, one multiply and four additions suffice for the three-port parallel junction with reflection-free port.
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