The Loaded $N$-Port Scattering Junction

Four Ideal Strings Intersecting at a Load

Series junction $\Leftrightarrow$ common velocity, forces sum to 0:

$$\begin{eqnarray*} V_1(s) = V_2(s) = \cdots = V_N(s) &\mathrel{\stackrel{\mathrm{... ... &=& V_J(s) R_J(s) \mathrel{\stackrel{\mathrm{\Delta}}{=}}F_J(s) \end{eqnarray*}$$

Computing common velocity at junction:

$$\begin{eqnarray*} R_J V_J &=& \sum_{i=1}^N F_i = \sum_{i=1}^N (F^+_i + F^-_i) \\... ...^-_i}_{V_J-V^+_i}) \\ &=& \sum_{i=1}^N (2 R_i V^+_i - R_i V_J) \end{eqnarray*}$$

$\Rightarrow$

$$V_J = 2\left(R_J + \sum_{i=1}^N R_i\right)^{-1} \sum_{i=1}^N R_i V^+_i$$

or

$$\zbox{V_J = \sum_{i=1}^N{\cal A}_i(s) V^+_i(s)}$$

where

$$\zbox{{\cal A}_i(s) \mathrel{\stackrel{\mathrm{\Delta}}{=}}\frac{2R_i}{R_J(s) + R_1 + \cdots + R_N}}$$

(generalized alpha parameter)

Finally, by continuity, $V_J = V_i = V^+_i + V^-_i\;\Rightarrow$

$$\zbox{V^-_i(s) = V_J(s) - V^+_i(s)}$$

Remarks:

• Lossless only when Re$\{R_J(j\omega)\}\equiv0$
• Memoryless only when Im$\{R_J(j\omega)\}\equiv0$
• Dynamic load takes all scattering coefficients into Laplace domain
  • Order of each ``scattering-filter'' equals load order
  • Normally one filter can serve entire junction
• Junction load equivalent to an $N+1$st waveguide having (generalized) wave impedance given by load impedance
  (consider ``perfect termination'' of a transmission line using a resistor equal in value to line impedance)