General Series Adaptor for Force Waves
In the more general case of ports being connected in
series, we have the physical constraints

The derivation is the dual of that in the parallel case (cf.
Eq.(F.16)), i.e., force and velocity are interchanged, and impedance
and admittance are interchanged:

The outgoing wave variables are given by

Beta Parameters
It is customary in the wave digital filter literature to define the beta parameters as
where

However, we normally employ a mixture of parallel and series adaptors,
while keeping a force-wave simulation. Since
, we obtain, after a small amount of algebra, the following
recipe for the series force-wave adaptor:
We see that we have




Reflection Coefficient, Series Case
The velocity reflection coefficient seen at port
is defined as
Representing the outgoing velocity wave



where






Equating like terms with Eq.(F.32) gives
Thus, the





Physical Derivation of Series Reflection Coefficient
Physically, the force-wave reflection coefficient seen at port
of a series adaptor is due to an impedance step from
, that
of the port interface, to a new impedance consisting of the series
combination of all other port impedances meeting at the
junction. Let
denote this series combination. Then we must have, as in Eq.

![]() |
(F.36) |
Let's check this ``physical'' derivation against the formal definition
Eq.(F.31) leading to
in Eq.
(F.33).
Define the total junction impedance as






Since

Series Reflection Free Port
For port to be reflection free in a series adaptor, we require
That is, the port's impedance must equal the series combination of the other port impedances at the junction. This result can be compared with that for the parallel junction in §F.2.2.
The series adaptor has now been derived in a way which emphasizes its duality with respect to the parallel adaptor.
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Two-Port Series Adaptor for Force Waves