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General Parallel Adaptor for Force Waves

In the more general case of $ N$ wave digital element ports being connected in parallel, we have the physical constraints

    $\displaystyle f_1(n) = f_2(n) = \cdots = f_N(n) \isdef f_J(n)$ (F.11)
    $\displaystyle v_1(n) + v_2(n) + \cdots + v_N(n) = 0$ (F.12)

The derivation for the two-port case extends to the $ N$ -port case without modification:
0 $\displaystyle =$ $\displaystyle \sum_{i=1}^N v_i$  
  $\displaystyle =$ $\displaystyle \sum_{i=1}^N\frac{f^{{+}}_i-f^{{-}}_i}{R_i}$  
  $\displaystyle =$ $\displaystyle \sum_{i=1}^N\frac{2f^{{+}}_i-f_J}{R_i}$  
  $\displaystyle \isdef$ $\displaystyle \sum_{i=1}^N \left(2\Gamma _if^{{+}}_i-\Gamma _i f_J \right)$  
$\displaystyle \,\,\Rightarrow\,\,
\sum_{j=1}^N \Gamma _j f_J$ $\displaystyle =$ $\displaystyle \sum_{i=1}^N 2\Gamma _i f^{{+}}_i$  
$\displaystyle \,\,\Rightarrow\,\,
f_J$ $\displaystyle =$ $\displaystyle \frac{\sum_{i=1}^N 2\Gamma _i f^{{+}}_i}{\sum_{j=1}^N \Gamma _j} .
\protect$ (F.13)

The outgoing wave variables are given by

$\displaystyle f^{{-}}_i(n) = f_J(n) - f^{{+}}_i(n)
$



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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2015-05-22 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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