Two-Port Series Adaptor for Force Waves

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

Figure N.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.

As discussed in §J.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 §N.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. (N.14) to get

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

which agrees as it must with the series velocity scattering junction relations in (J.12) and diagrammed in Fig. J.13. 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. N.6b. The one-multiply form is now

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