Solution Properties

We determined that

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

where

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

• 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 (with no input) having (generalized) wave impedance given by load impedance $R_J(s)$
  (consider ``perfect termination'' of a transmission line using a resistor equal in value to line impedance)