As we saw for the parallel case in §F.2.3, the series three-port adaptor (scattering junction) with reflection-free port can be implemented using only one multiply.
The scattering relations in terms of the beta parameters introduced in §F.2.5 for a series junction
of any number of ports:
and is the wave impedance on port .
As for the parallel case, we set , , , and . With port A reflection-free, we have . Dividing all impedances by gives , and . Our one degree of freedom may be chosen as , and then .
The beta parameters become
, so that the scattering formulas are
The series adaptor has now been derived in a way which emphasizes its duality with respect to the parallel adaptor.