General Series Adaptor for Force Waves

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General Series Adaptor for Force Waves

In the more general case of 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. $\,$ (Q.16)), i.e., force and velocity are interchanged, and impedance and admittance are interchanged:

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

The outgoing wave variables are given by

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

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Chapters

Physical Audio Signal Processing
    Wave Digital Filters
       Adaptors for Wave Digital Elements
          General Series Adaptor for Force Waves