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Alpha Parameters

Recall $ N$ velocity waveguides meeting at a series junction:

\begin{eqnarray*}
V_J(s) &=& \sum_{i=1}^N{\cal A}_i(s) V^+_i(s)\qquad\mbox{(junction velocity)}\\ [10pt]
V^-_i(s) &=& V_J(s) - V^+_i(s)\qquad\;\mbox{(outgoing velocity waves)}
\end{eqnarray*}

where

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

In the lossless (unloaded) case, $ R_J(s)=0$ , and so the alpha parameters are real and positive and add up to 2:

$\displaystyle \alpha_i = \frac{2R_i}{R_1 + \cdots + R_N}
$

I.e.,

$\displaystyle 0\leq\alpha_i \leq 2
$

and

$\displaystyle \sum_{i=1}^N\alpha_i = 2
$

Also, we no longer have to be in the Laplace domain ($ R_J=0$ )



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``Scattering at an Impedance Discontinuity'', by Julius O. Smith III, (From Lecture Overheads, Music 420).
Copyright © 2014-03-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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