### 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 are the port impedances (attached element reference impedances). In terms of the beta parameters, the force-wave series adaptor performs the following computations:

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 multiplies and additions as in the parallel-adaptor case. However, we again have from Eq.(F.26) that

*dependent port*.

#### Reflection Coefficient, Series Case

The *velocity reflection coefficient* seen at port
is defined as

Representing the outgoing velocity wave as the superposition of the reflected wave plus the transmitted waves from the other ports, we have

where denotes the

*velocity transmission coefficient*

*velocity!transmission coefficient*from port to port . Substituting Eq.(F.29) into Eq.(F.30) yields

Equating like terms with Eq.(F.32) gives

Thus, the th beta parameter is the velocity transmission coefficient from th port to any other port (besides the th). To convert the transmission coefficient from the th port to the reflection coefficient for that port, we simply subtract 1. These relationships are specific to velocity waves at a series junction (cf. Eq.(F.22)). They are exactly the dual of Equations (F.22-F.23) for force waves at a parallel junction.

#### 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.25),

(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

*all*impedances connected to the junction. Then by Eq.(F.35), for all . From Eq.(F.26), the

*velocity*reflection coefficient is given by

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