### 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

 (F.26)

where are the port impedances (attached element reference impedances). In terms of the beta parameters, the force-wave series adaptor performs the following computations:
 (F.27) (F.28)

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:

 (F.29) (F.30)

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

so that we may implement one beta parameter as 2 minus the sum of the rest, thus eliminating a multiplication by creating a dependent port.

#### Reflection Coefficient, Series Case

The velocity reflection coefficient seen at port is defined as

 (F.31)

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

 (F.32)

where denotes the velocity transmission coefficientvelocity!transmission coefficient from port to port . Substituting Eq.(F.29) into Eq.(F.30) yields

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

 (F.33) (F.34)

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

 (F.35)

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

This is the series combination of 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

the result follows.

#### Series Reflection Free Port

For port to be reflection free in a series adaptor, we require

 (F.37)

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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