One-Multiply Parallel Reflection-Free Three-Port Adaptor

It turns out only one multiply is needed to implement three-port adaptors (scattering junctions) having a reflection-free port. The derivation is very similar to deriving the one-multiply two-port scattering junction from the four-multiply Kelly-Lochbaum scattering junction, as discussed in Fig..

Let's begin with the scattering relations in terms of the alpha parameters introduced in §F.2.2 for a parallel junction of any number of ports:

$$f_J = \sum_{i=1}^N \alpha_i \, f^{{+}}_i,$$ (F.23)

$$f^{{-}}_i(n) = f_J(n) - f^{{+}}_i(n)$$ (F.24)

where $f_J$ denotes the junction force (or voltage), the $\alpha$ parameters are defined for parallel junctions by

$$\alpha_i \isdefs \frac{2\Gamma _i}{\sum_{j=1}^N \Gamma _j},$$

and $\Gamma _i=1/R_i$ is the wave admittance on port $i$ .

Here we wish to focus on the three-port case $N=3$ . To connect with our previous example, set $R_1=R_A$ , $R_2=R_B$ , and $R_3=R_C$ . When port A is reflection-free, we have the constraint that $\Gamma _A=\Gamma _B+\Gamma _B$ ( $R_A=R_B\Vert R_C=R_BR_C/(R_B+R_C)$ ). Scattering relations are unchanged when all port impedances are divided by the same positive number, so divide through by $R_A$ to obtain $R_1=\Gamma _1=1$ , and $\Gamma _2+\Gamma _3=1$ . With $\Gamma _1=1$ , we have $\Gamma _2\in[0,1]$ , and $\Gamma _3=1-\Gamma _2$ . In other words, there is only one degree of freedom, $\Gamma _2\in[0,1]$ for the three-port parallel junction with a reflection-free port.

The alpha parameters become $\alpha_1=1$ , $\alpha_2=\Gamma _2\in[0,1]$ , and $\alpha_3=\Gamma _3=1-\Gamma _2\in[0,1]$ , so that the scattering formulas simplify to

$$f_J = f^{{+}}_1 + \Gamma _2f^{{+}}_2 + (1-\Gamma _2)f^{{+}}_3 \eqsp f^{{+}}_1 + f^{{+}}_3 + \Gamma _2(f^{{+}}_2 - f^{{+}}_3),$$ (F.25)

$$f^{{-}}_i = f_J - f^{{+}}_i.$$ (F.26)

Thus, only one multiply is necessary to compute the junction force (or voltage). In this form we have six additions, but this can be brought down to four. Expand $f_J$ in the last equation to get

$$f_\delta \isdef f^{{+}}_2 - f^{{+}}_3$$ (F.27)

$$g_\delta \isdef \Gamma _2 f_\delta$$ (F.28)

$$f^{{-}}_1 = f^{{+}}_3 + g_\delta$$ (F.29)

$$f^{{-}}_3 = f^{{+}}_1 + g_\delta$$ (F.30)

$$f^{{-}}_2 = f^{{+}}_1 + f^{{+}}_3 + g_\delta - f^{{+}}_2 \eqsp f^{{+}}_1 + g_\delta - f_\delta$$ (F.31)

$$= f^{{-}}_3 - f_\delta$$ (F.32)

Thus, one multiply and four additions suffice for the three-port parallel junction with reflection-free port.