General Parallel Adaptor for Force Waves

In the more general case of $N$ wave digital element ports being connected in parallel, we have the physical constraints

$$f_1(n) = f_2(n) = \cdots = f_N(n) \isdef f_J(n)$$ (P.14)

$$v_1(n) + v_2(n) + \cdots + v_N(n) = 0$$ (P.15)

The derivation for the two-port case extends to the $N$-port case without modification:

$$= \sum_{i=1}^N v_i$$ 0

$$= \sum_{i=1}^N\frac{f^{{+}}_i-f^{{-}}_i}{R_i}$$

$$= \sum_{i=1}^N\frac{2f^{{+}}_i-f_J}{R_i}$$

$$\isdef \sum_{i=1}^N \left(2\Gamma _if^{{+}}_i-\Gamma _i f_J \right)$$

$$\,\,\Rightarrow\,\, \sum_{j=1}^N \Gamma _j f_J = \sum_{i=1}^N 2\Gamma _i f^{{+}}_i$$

$$\,\,\Rightarrow\,\, f_J = \frac{\sum_{i=1}^N 2\Gamma _i f^{{+}}_i}{\sum_{j=1}^N \Gamma _j} . \protect$$ (P.16)

The outgoing wave variables are given by

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