Wave Variables

• In order to circumvent the problem of delay-free loops, Fettweis introduced $wave$ $variables$:
  $$\begin{eqnarray*} a_{n} &=& v_{n}+i_{n}R_{0}\\ b_{n} &=& v_{n}-i_{n}R_{0} \end{eqnarray*}$$
  

  where $R_{0}$ is an arbitrary parameter called a $port$ $resistance$ (for reasons we'll discuss later).

• In matrix notation, we readily see that this is a linear transformation of the state variables $\{v_n,i_n\}$:
  $$\left[\begin{array}{c} a_n \\ [2pt] b_n \end{array}\right] = \lef... ... \end{array}\right] \left[\begin{array}{c} v_n \\ [2pt] i_n \end{array}\right]$$

• Since the determinant of the two-by-two matrix is $-2R_0$, the transformation is non-singular provided $R_0\neq 0$. We will additionally stipulate $R_0>0$.

• The inverse transformation is
  $$\begin{eqnarray*} v_n &=& \frac{a_n+b_n}{2} \\ i_n &=& \frac{a_n-b_n}{2R_0} \end{eqnarray*}$$