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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

 (2.31) (2.32)

where we recall from (2.15) that is defined as the reciprocal of the port resistance . For power-normalized wave variables, we thus have, applying (2.17),

 (2.33) (2.34)

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 three-port adaptors, for either voltage or power-normalized waves, are shown in Figure 2.12.

A useful simplification occurs when we can choose, for a particular port (called a reflection-free port [57]) of an -port adaptor,

in which case the scattering equations (2.31) yield, for the output wave at port ,

Thus, at a reflection-free port , the output wave is independent of the input wave ; such a port can be connected to any other without risk of a resulting delay-free loop. The same choices of port resistances (2.35) will also give a reflection-free port if power wave variables are employed.

Subsections

Next: Scattering Matrices for Adaptors Up: Wave Digital Elements and Previous: The Unit Element
Stefan Bilbao 2002-01-22