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Two-Port Series Adaptor for Force Waves

Figure F.6a illustrates a generic two-port description of the series adaptor.

Figure F.6: a) Two-port description of the adaptor implementing a series connection between reference impedances $ R_1$ and $ R_2$ . b) Corresponding series force scattering junction (adaptor wave flow diagram) in Kelly-Lochbaum form.
\includegraphics[width=\twidth]{eps/lAdaptorSeries}

As discussed in §7.2, a series connection is characterized by a common velocity and forces which sum to zero at the junction:

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

The derivation can proceed exactly as for the parallel junction in §F.2.1, but with force and velocity interchanged, i.e., $ f\leftrightarrow v$ , and with impedance and admittance interchanged, i.e., $ R\leftrightarrow \Gamma $ . In this way, we may take the dual of Eq.$ \,$ (F.11) to get

\begin{eqnarray*}
v^{-}_1 &=& -\rho v^{+}_1 + (1+\rho) v^{+}_2\\
v^{-}_2 &=& (1-\rho)v^{+}_1 + \rho v^{+}_2
\end{eqnarray*}

diagrammed in Fig.F.7. Converting back to force wave variables via $ f^{{+}}_i=R_iv^{+}_i$ and $ f^{{-}}_i=-R_iv^{-}_i$ , and noting that $ (1+\rho)R_1/R_2 = 1-\rho$ , we obtain, finally,

\begin{eqnarray*}
f^{{-}}_1 &=& \rho f^{{+}}_1 - (1-\rho) f^{{+}}_2\\
f^{{-}}_2 &=& -(1+\rho)f^{{+}}_1 - \rho f^{{+}}_2
\end{eqnarray*}

as diagrammed in Fig.F.6b. The one-multiply form is now

\begin{eqnarray*}
f^{{-}}_1 &=& -f^{{+}}_2 + \rho(f^{{+}}_1 + f^{{+}}_2)\\
f^{{-}}_2 &=& -f^{{+}}_1 - \rho(f^{{+}}_1 + f^{{+}}_2).
\end{eqnarray*}

Figure F.7: Series velocity scattering junction in Kelly-Lochbaum form.
\includegraphics[scale=0.9]{eps/lscat_vel_series_renum}


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-06-11 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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