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The Loaded $ N$ -Port Scattering Junction

Four Ideal Strings Intersecting at a Load

\epsfig{file=eps/fNstrings.eps,width=1.1\textwidth }

Series junction $ \Leftrightarrow$ common velocity, forces sum to 0:

\begin{eqnarray*}
V_1(s) = V_2(s) = \cdots = V_N(s) &\mathrel{\stackrel{\mathrm{\Delta}}{=}}& V_J(s) \\
F_1(s) + F_2(s) + \cdots + F_N(s) &=& V_J(s) R_J(s) \mathrel{\stackrel{\mathrm{\Delta}}{=}}F_J(s)
\end{eqnarray*}

Computing common velocity at junction:

\begin{eqnarray*}
R_J V_J \;=\; F_j &=& \sum_{i=1}^N F_i = \sum_{i=1}^N (F^+_i + F^-_i) \\
&=& \sum_{i=1}^N (R_i V^+_i - R_i \underbrace{V^-_i}_{V_J-V^+_i}) \\
&=& \sum_{i=1}^N (2 R_i V^+_i - R_i V_J)
\end{eqnarray*}

$ \Rightarrow$

$\displaystyle V_J = 2\left(R_J + \sum_{i=1}^N R_i\right)^{-1} \sum_{i=1}^N R_i V^+_i
$

or

$\displaystyle \zbox{V_J(s) = \sum_{i=1}^N{\cal A}_i(s) V^+_i(s)}
$

where

$\displaystyle \zbox{{\cal A}_i(s) \mathrel{\stackrel{\mathrm{\Delta}}{=}}\frac{2R_i}{R_J(s) + R_1 + \cdots + R_N}}
$

(generalized ``alpha parameter'', cf. Wave Digital Filters)

Finally, by continuity, $ V_J = V_i = V^+_i + V^-_i\;\Rightarrow$

$\displaystyle \zbox{V^-_i(s) = V_J(s) - V^+_i(s)}
$



Subsections
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``Scattering at an Impedance Discontinuity'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2015-05-19 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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