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The (2+1)D Parallel-plate System

Generalizing the above procedure to several dimensions is straightforward. We examine here, as a practical example, the (2+1)D parallel-plate system, which is written as:
$\displaystyle \begin{eqnarray}l\frac{\partial i_{x}}{\partial t}+\frac{\partial...
...al i_{x}}{\partial x}+\frac{\partial i_{y}}{\partial y}+gu+h&=&0 \end{eqnarray}$ (3.63a)

This system was treated using MDWDFs in [62,211]. Now the dependent variables are a voltage $ u$, and current density components $ i_{x}$ and $ i_{y}$; these, and the sources $ e$, $ f$ and $ h$ are functions of time $ t$ and two spatial variables, $ x$ and $ y$. $ l$, $ c$, $ r$ and $ g$ are arbitrary smooth positive functions of $ x$ and $ y$ ($ l$ and $ c$ are strictly positive). It is worth mentioning that the same equations can be used in the contexts of (2+1)D linear acoustics, the vibration of a membrane, and, with a trivial modification, (2+1)D electromagnetic field problems (involving TE or TM modes).

System (3.67) is symmetric hyperbolic, and thus has the form of (3.1), where $ {\bf w} = [i_{x}, i_{y}, u]^{T}$, and with

$\displaystyle {\bf P} = \begin{bmatrix}l&0&0\\ 0&l&0\\ 0&0&c\\ \end{bmatrix}\hs...
...\ \end{bmatrix}\hspace{0.2in}{\bf f} = \begin{bmatrix}e\\ f\\ h\\ \end{bmatrix}$    

It will follow, as in the case of the (1+1)D transmission line system (see §3.7.4), that the total energy of the MDKC that we will derive in the next section will be equal to the energy of system (3.67), as per (3.6).



Subsections
next up previous
Next: Phase and Group Velocity Up: Multidimensional Wave Digital Filters Previous: Simplified Networks
Stefan Bilbao 2002-01-22