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Adaptors

Consider now a series connection of $ M$ ports, where we have a port resistance $ R_{j}>0$, $ j=1,\hdots,M$, associated with each port. In terms of instantaneous quantities, we have

$\displaystyle \sum_{j=1}^{M}v_{j} = 0$    

or, in terms of wave variables, using the inverse of the transformation (2.14),

$\displaystyle \sum_{j=1}^{M}\left(a_{j}+b_{j}\right) = 0$    

Since the currents at all ports are all equal to $ i$, this implies, using $ b_{j} = a_{j}-2R_{j}i$, that

$\displaystyle \sum_{j=1}^{M}\left(-2R_{j}i+2a_{j}\right) = 0$    

and thus

$\displaystyle i = \frac{1}{\sum_{j=1}^{M}R_{j}}\sum_{j=1}^{M}a_{j}$ (2.30)

By applying similar manipulations in the case of a parallel connection of $ M$ ports, we can then write down the equations relating the input and output wave variables at the $ k$th port for both types of connection as

$\displaystyle b_{k}$ $\displaystyle = a_{k} -\frac{2R_{k}}{\sum_{j=1}^{M}R_{j}}\sum_{j=1}^{M}a_{j},$   $\displaystyle k=1,\hdots,M$   $\displaystyle {\mbox {\rm Series connection}}$ (2.31)
$\displaystyle b_{k}$ $\displaystyle = -a_{k} +\frac{2}{\sum_{j=1}^{M}G_{j}}\sum_{j=1}^{M}G_{j}a_{j},$   $\displaystyle k=1,\hdots,M$   $\displaystyle {\mbox {\rm Parallel connection}}$ (2.32)

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

$\displaystyle \underline{b}_{k}$ $\displaystyle = \underline{a}_{k} -\frac{2\sqrt{R_{k}}}{\sum_{j=1}^{M}R_{j}}\sum_{j=1}^{M}\sqrt{R_{j}}\underline{a}_{j},$   $\displaystyle k=1,\hdots,M$   $\displaystyle {\mbox {\rm Series connection}}$ (2.33)
$\displaystyle \underline{b}_{k}$ $\displaystyle = -\underline{a}_{k} +\frac{2\sqrt{G_{k}}}{\sum_{j=1}^{M}G_{j}}\sum_{j=1}^{M}\sqrt{G_{j}}\underline{a}_{j},$   $\displaystyle k=1,\hdots,M$   $\displaystyle {\mbox {\rm Parallel connection}}$ (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.

Figure 2.12: Three-port adaptors-- (a) a general three-port series adaptor and one for which port 3 is reflection-free and (b) a general three-port parallel adaptor and one for which port 3 is reflection-free.
\begin{figure}\begin{center}
\begin{picture}(490,130)
\par % graphpaper(0,0)(49...
...(a)}
\put(399,-20){(b)}
\end{picture} \end{center} \vspace{0.1in}
\end{figure}

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

\begin{subequations}\begin{align}R_{q} &= \sum_{j=1, j\neq q}^{M}R_{j}&&{\mbox {...
...x {\rm Parallel reflection-free port conductance}} \end{align}\end{subequations}

in which case the scattering equations (2.31) yield, for the output wave at port $ q$,
\begin{subequations}\begin{align}b_{q} &= -\sum_{j=1, j\neq q}^{M}a_{j} &&{\mbox...
..._{j} &&{\mbox {\rm Parallel reflection-free port}} \end{align}\end{subequations}

Thus, at a reflection-free port $ q$, the output wave $ b_{q}$ is independent of the input wave $ a_{q}$; 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 up previous
Next: Scattering Matrices for Adaptors Up: Wave Digital Elements and Previous: The Unit Element
Stefan Bilbao 2002-01-22