Next: The Waveguide Mesh Up: The (2+1)D Parallel-plate System Previous: The (2+1)D Parallel-plate System

## Defining Equations and Centered Differences

The set of PDEs describing a lossless, source-free parallel-plate transmission line in (2+1)D is a direct generalization of system (4.17):
 (4.60a)

Now and are the components of the current density vector in the and directions, respectively, and is the voltage between the plates. and , both assumed positive everywhere, are the inductance and capacitance per unit length.

If we assume that and are constant, then as in the (1+1)D case, the set of equations can be reduced to a single second order equation in the voltage alone:

 (4.61)

and again, the wave speed is given by

The centered difference scheme for system (4.49) also generalizes simply. Define grid functions , and which run over half-integer values of , , and , i.e.,

We will furthermore assume that the spatial step in the direction and the direction are the same and equal to . As before, the time step will be . We can use the approximations (4.19), as well as an approximation to the derivative in the direction,

where stands for either of or . We obtain the difference scheme

where we have written

and

As in the (1+1)D case, it is possible to subdivide the calculation scheme (4.51) into smaller, mutually exclusive subschemes. Using a decimated grid for the variable coefficient difference scheme amounts to rewriting scheme (4.51) as

where we now compute solutions for , and integer. The interleaved grid is shown in Figure 4.18; a grey (white) dot at a grid location indicates that the adjacent named variable is to be calculated at times which are even (odd) multiples of . This interleaved form was originally put forth by Yee [214] in the context of electromagnetic field problems, and forms the basis of the widely used finite-difference time domain (FDTD) family of difference methods [184], which were discussed briefly in §4.1. If system (4.52) is rewritten as a TE or TM system, the interleaved arrangement of the field components also has an interesting physical interpretation as a discrete counterpart to the integral form of Ampere's and Faraday's Laws [184]. This result also extends easily to the discretization of Maxwell's equations in (3+1)D [214]; see §4.10.6 for more details.

The (2+1)D analogue of (4.23), which holds in the case where and are constant, is

 (4.64) (4.65)

The magic time step will now be

and (4.53) simplifies to

 (4.66)

As in (1+1)D, when we are solving the wave equation by centered differences at the magic time step (or at CFL), the calculation further decomposes into two independent calculations; we need only update for even (or odd). We will examine this interesting decomposition property in detail in Appendix A.

Next: The Waveguide Mesh Up: The (2+1)D Parallel-plate System Previous: The (2+1)D Parallel-plate System
Stefan Bilbao 2002-01-22