Energetic Interpretation

Let us now reexamine the passive MDKC in Figure 3.14(a). The total stored energy flux in the network is contained in the four inductors and will be, from (3.35),

$${\bf E}_{total} = \frac{1}{2}L_{1}i_{1}^{2}{\bf e}_{t'}+\frac{1}{... ...}L_{0}(i_{1}+i_{2})^{2}{\bf e}_{1}+\frac{1}{2}L_{0}(i_{1}-i_{2})^{2}{\bf e}_{2}$$

where the ${\bf e}_{t'}$ is a unit vector in direction $t'$, and ${\bf e}_{1}$ and ${\bf e}_{2}$ are unit vectors in directions $t_{1}$ and $t_{2}$ respectively. Applying the definitions of the inductances, from (3.63), the current definitions from (3.60), and the fact that ${\bf e}_{1} = \left({\bf e}_{t'}+{\bf e}_{x}\right)/\sqrt{2}$ and ${\bf e}_{2} = \left({\bf e}_{t'}-{\bf e}_{x}\right)/\sqrt{2}$, then this total energy flux can be rewritten as

$${\bf E}_{total} = \frac{1}{2}v_{0}li^{2}{\bf e}_{t'}+\frac{1}{2}v... ...e}_{x}= \frac{1}{2}li^{2}{\bf e}_{t}+\frac{1}{2}cu^{2}{\bf e}_{t}+ui{\bf e}_{x}$$

Here ${\bf e}_{t} = v_{0}{\bf e}_{t'}$ is a unit vector in the time direction. The total scalar energy at time $t$ of this network will, from (3.30), be

$$\mathcal{E}(t) = \int_{-\infty}^{+\infty}{\bf e}_{t}^{T}{\bf E}_{... ...u^{2}\right)dx = \int_{-\infty}^{+\infty}\frac{1}{2}{\bf w}^{T}{\bf P}{\bf w}dx$$

and thus coincides with the energy definition of the symmetric hyperbolic system, from (3.6). This is certainly not surprising, but the important point here is that in an MDKC such as that of Figure 3.14(a), the scalar energy has been broken down into contributions from several interacting components (the inductors), each of which is passive individually; this useful energy subdivision has been exploited here as a means of developing passive numerical methods.