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MDKC and MDWD Network
The circuit can be derived along the same lines as for the (1+1)D case; we deal here with the discretization on a rectilinear grid, and will thus apply coordinate transformation defined by the of (3.22). Rewriting system (3.67) in terms of the new coordinates
using
, with the pseudo inverse (3.23) gives
where we have used the new currentlike variables
and is, as in the (1+1)D case, an arbitrary positive constant (which has also been used to scale (3.69c)).
will be treated as a simple time derivative, according to the generalized trapezoid rule discussed in §3.5.1. Figure 3.17 shows the MDKC that results from the transformed set of equations (3.69). The MDWD network corresponding to the MDKC is shown in Figure 3.18, where we have used stepsizes
,
.
Figure 3.18:
MDWD network for the (2+1)D parallelplate system, in rectangular coordinates.

Passivity follows from a positivity condition on the network inductances, in particular , and (the values of which are given in Figure 3.17). These conditions are

(3.66) 
The choice of
, where
and
gives a stability bound of

(3.67) 
which is the best possible bound for this network [61]. Note that is again bounded away from the maximum group velocity, even taking into account the scaling factor ( in this case), which is a consistent feature of explicit numerical methods in multiple spatial dimensions.
If and are constant, and in addition , , , and are zero, and (3.71) holds with equality, i.e., we have

(3.68) 
then the network of Figure 3.18 simplifies to the structure shown in Figure 3.19. This particular structure bears a very strong resemblance to the (2+1)D waveguide mesh [157,198] which we saw briefly in §1.1.2, and will examine in detail in Chapter 4.
Figure 3.19:
Simplified MDWD network for the (2+1)D transmission line equations, in the lossless, sourcefree and constant parameter case.

Next: Finite Difference Interpretation
Up: The (2+1)D Parallelplate System
Previous: Phase and Group Velocity
Stefan Bilbao
20020122