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Boundary Conditions
We now examine the termination of the waveguide mesh which simulates the behavior of the (2+1)D parallel-plate equations. The two most important types of boundary conditions are
![$\displaystyle u$](img1639.png) |
![$\displaystyle =0$](img250.png) |
|
Short-circuit termination |
(4.80) |
![$\displaystyle i_{n}$](img1640.png) |
![$\displaystyle =0$](img250.png) |
|
Open-circuit termination |
(4.81) |
where
refers to the component of
which is normal to the boundary. Condition (4.68) corresponds to a transmission line plate pair which are connected (and thus short-circuited) at the boundary; the same condition holds for a clamped membrane for which
is interpreted as a transverse velocity, and
as in-plane forces. Condition (4.69) is an open-circuited termination; current can not leave the plate at the edges. This second condition is analogous to the rigid termination of a (2+1)D acoustic medium, where
are interpreted as flow velocities, and
as a pressure. Both conditions are of the form of (3.8), and are lossless. We will examine only the termination of the mesh on a rectangular domain (though the result extends easily to the radial mesh to be discussed in §4.6.2).
In the case of the (1+1)D transmission line, we could treat a staggered mesh terminated by a parallel junction, and through the duality of
and
extend the result to include termination by a series junction (see §4.3.9). This is no longer the case in (2+1)D, and we must treat the two types of termination separately. Consider a bottom (southern) boundary at
of an interleaved mesh of the type shown in Figure 4.21. The two possible types of termination are shown in Figure 4.23.
Figure 4.23:
Grid terminations at a southern boundary.
![\begin{figure}\begin{center}
\begin{picture}(500,100)
% graphpaper(0,0)(500,100...
...(b)}
\put(232,5){$y=0$}
\end{picture} \vspace{0.2in}
\end{center} \end{figure}](img1643.png) |
Subsections
Next: Grid Arrangement Requiring Voltage
Up: The (2+1)D Parallel-plate System
Previous: Reduced Computational Complexity and
Stefan Bilbao
2002-01-22