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

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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