Alpha Parameters

Recall $N$ velocity waveguides meeting at a series junction:

$$\begin{eqnarray*} V_J(s) &=& \sum_{i=1}^N{\cal A}_i(s) V^+_i(s)\qquad\mbox{(junction velocity)}\\ [10pt] V^-_i(s) &=& V_J(s) - V^+_i(s)\qquad\;\mbox{(outgoing velocity waves)} \end{eqnarray*}$$

where

$${\cal A}_i(s) \isdef \frac{2R_i}{R_J(s) + R_1 + \cdots + R_N}$$

In the lossless (unloaded) case, $R_J(s)=0$ , and so the alpha parameters are real and positive and add up to 2:

$$\alpha_i = \frac{2R_i}{R_1 + \cdots + R_N}$$

I.e.,

$$0\leq\alpha_i \leq 2$$

and

$$\sum_{i=1}^N\alpha_i = 2$$

Also, we no longer have to be in the Laplace domain ($R_J=0$ )