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:
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(F.45) |
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(F.46) |
and
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
,
, and
, so that the scattering formulas are
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(F.47) |
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(F.48) |
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(F.49) |
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(F.50) |
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(F.51) |
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(F.52) |
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(F.53) |
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(F.54) |
The series adaptor has now been derived in a way which emphasizes its duality with respect to the parallel adaptor.