Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

General Series Adaptor for Force Waves

In the more general case of $ N$ ports being connected in series, we have the physical constraints

\begin{eqnarray*}
&& v_1(n) = v_2(n) = \cdots = v_N(n) \isdef v_J(n)\\
&& f_1(n) + f_2(n) + \cdots + f_N(n) = 0
\end{eqnarray*}

The derivation is the dual of that in the parallel case (cf. Eq.$ \,$ (F.13)), i.e., force and velocity are interchanged, and impedance and admittance are interchanged:

\begin{eqnarray*}
0 &=& \sum_{i=1}^N f_i \\
&=& \sum_{i=1}^NR_i\left(v^{+}_i-v^{-}_i\right)\\
&=& \sum_{i=1}^NR_i\left(2v^{+}_i-v_J\right)\\
\,\,\Rightarrow\,\,\quad
\sum_{j=1}^N R_j v_J &=& \sum_{i=1}^N 2R_i v^{+}_i\\
\,\,\Rightarrow\,\,\quad
v_J &=& \frac{\sum_{i=1}^N 2R_i v^{+}_i}{\sum_{j=1}^N R_j} .
\end{eqnarray*}

The outgoing wave variables are given by

$\displaystyle v^{-}_i(n) = v_J(n) - v^{+}_i(n)
$



Subsections
Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work]  [Order a printed hardcopy]  [Comment on this page via email]

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-10-15 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA