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Adaptors
Consider now a series connection of ports, where we have a port resistance ,
, associated with each port. In terms of instantaneous quantities, we have
or, in terms of wave variables, using the inverse of the transformation (2.14),
Since the currents at all ports are all equal to , this implies, using
, that
and thus

(2.30) 
By applying similar manipulations in the case of a parallel connection of ports, we can then write down the equations relating the input and output wave variables at the th port for both types of connection as
where we recall from (2.15) that is defined as the reciprocal of the port resistance .
For powernormalized wave variables, we thus have, applying (2.17),
The operator which performs this calculation on the wave variables is called a series adaptor or a parallel adaptor [46], depending on the type of connection. The graphical representations of threeport adaptors, for either voltage or powernormalized waves, are shown in Figure 2.12.
Figure 2.12:
Threeport adaptors (a) a general threeport series adaptor and one for which port 3 is reflectionfree and (b) a general threeport parallel adaptor and one for which port 3 is reflectionfree.

A useful simplification occurs when we can choose, for a particular port (called a reflectionfree port [57]) of an port adaptor,
in which case the scattering equations (2.31) yield, for the output wave at port ,
Thus, at a reflectionfree port , the output wave is independent of the input wave ; such a port can be connected to any other without risk of a resulting delayfree loop. The same choices of port resistances (2.35) will also give a reflectionfree port if power wave variables are employed.
Subsections
Next: Scattering Matrices for Adaptors
Up: Wave Digital Elements and
Previous: The Unit Element
Stefan Bilbao
20020122