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The Lossless Junction

A scattering junction of waveguide sections is characterized by its scattering matrix ${\mbox{\boldmath$A$}}$. The relationship between the $N$ incoming and outgoing traveling waves is given by:

\end{displaymath} (23)

where ${\mbox{\boldmath$p$}}^+$ is the vector of incoming waves (assumed scalar here) and ${\mbox{\boldmath$p$}}^-$ is the vector of outgoing waves relative to the junction (see Fig. 2). We say that the junction is $N$-way (or it has $N$ branches) if $N$ is the dimension of the incoming and outgoing wave vectors.
Figure: A schematic depiction of the 3-way waveguide junction.
\begin{figure}\centerline{\epsfxsize=240pt \epsfbox{figure/junction3.eps}}\end{figure}

We now consider the case of a constant scattering matrix ${\mbox{\boldmath$A$}}$. The more general case of scattering matrices as functions of $z$ will be considered in Section 10.

The net complex power entering the junctions is

$\displaystyle {\cal P}= {\mbox{\boldmath$u$}}^* {\mbox{\boldmath$p$}}$ $\textstyle =$ $\displaystyle {\mbox{\boldmath$p$}}^{+*}{\mbox{\boldmath$\Gamma$}}^*(1/z^*) {\m...
...+- {\mbox{\boldmath$p$}}^{-*}{\mbox{\boldmath$\Gamma$}}{\mbox{\boldmath$p$}}^-+$  
    $\displaystyle {\mbox{\boldmath$p$}}^{+*}{\mbox{\boldmath$\Gamma$}}^*(1/z^*) {\m...
...^-- {\mbox{\boldmath$p$}}^{-*}{\mbox{\boldmath$\Gamma$}}{\mbox{\boldmath$p$}}^+$  
  $\textstyle =$ $\displaystyle ({\cal P}^+- {\cal P}^-) + ({\cal P}^\times - {\cal P}^{\times*})$ (24)

where ${\mbox{\boldmath$\Gamma$}}$ is the diagonal matrix containing the $N$ wave admittances of all the branches meeting at the junction. Assuming the branch admittances are Hermitian and nonzero, we have that ${\mbox{\boldmath$\Gamma$}}$ has positive real elements along its diagonal and zeros elsewhere. The quantity ${\cal P}^+= {\mbox{\boldmath$p$}}^{+*}{\mbox{\boldmath$\Gamma$}}{\mbox{\boldmath$p$}}^+\geq 0$ is incoming active power, and ${\cal P}^-= {\mbox{\boldmath$p$}}^{-*}{\mbox{\boldmath$\Gamma$}}{\mbox{\boldmath$p$}}^-\geq 0$ is then the outgoing active power relative to the junction. The term ${\cal P}^+- {\cal P}^-$ is the absorbed active power, while the term ${\cal P}^\times - {\cal P}^{\times*}$, containing the mixed incoming and outgoing waves, is called the reactive power.

A scattering junction is said to be passive when the absorbed active power is nonnegative, i.e., when

{\cal P}^+\geq {\cal P}^-
\end{displaymath} (25)

for $\vert z\vert \geq 1$. In other terms, the outgoing active power does not exceed the incoming active power.

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Download wgj.pdf

``Aspects of Digital Waveguide Networks for Acoustic Modeling Applications'', by Julius O. Smith III and Davide Rocchesso , December 19, 1997, Web published at
Copyright © 2007-02-07 by Julius O. Smith III and Davide Rocchesso
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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