Two-Port Series Adaptor for Force Waves

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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 and . 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.14) 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,