Boundary Conditions

Let us now examine some simple lossless boundary conditions of the form of (3.8). Suppose that we would like the boundary condition for the transmission line to be that of an *open-circuit* at the left termination, so that we have

Open-circuit termination |

The wave digital approximation to the physical current at any grid point is calculated from the wave variables incident on the left series adaptor, so that we have

If we set, at the left-most grid point,

then the calculated current will be identically zero, and is easily obtained from and , both of which are available. A short-circuited termination, i.e.,

Short-circuit termination |

can be accomplished by treating the right-hand adaptor in a similar manner. It is possible to mix these conditions, and to introduce loss and a lumped terminating source as well [107].

This idea is also easily extended to multiple dimensions for rectilinearly sampled grids, if the boundary is parallel to one of the grid axes. We would like to add, however, that such a termination has *never been shown to be passive*, and that there is no general theory applicable to boundary termination of MDWD networks [142](though we will provide a possible foundation for such an approach in §6.2.3). While it is easy to prove that the simple cases above correspond to passive lumped terminations, there are situations in higher dimensions when this approach becomes difficult to apply reliably; in several instances, (see Chapter 5 for some added discussion), this approach has failed in simulation.
The difficulty with approaching boundary termination in this way is that the physics of the problem (in particular the passivity at the boundary) is not being taken into account; this method, though easy to apply, is essentially no different from what is done using conventional finite difference methods. Fettweis and Nitsche [61] provided an alternative method which is more satisfying from a physical point of view; in this case, the region beyond the boundary is modeled as a material with extreme parameter values (typically
or
, for the transmission line or parallel-plate problem). These regions are still passive, though it may now be necessary to employ a ``layer'' of this material, which will incur extra calculation costs.

Other recent work has involved more general lumped boundary terminations [5,211,212], as well as the termination of the (2+1)D parallel-plate problem in hexagonal coordinates; we mention that these approaches are unwieldy in the extreme; in at least one case [210], the proposed modelling of a passive boundary condition requires active elements!

The problem with the termination of MDWD networks is that when spatial dependence is expanded out to get a signal flow graph, we do not end up with a lumped network of portwise-connected elements; see, for example, the flow graph for the simple advective system, shown in Figure 3.7. Such is not the case for digital waveguide networks, which are in fact formulated from the outset as large lumped networks. For this reason, boundary termination is much simpler in a DWN. In Chapter 4, which is devoted to digital waveguide networks, we will discuss boundary termination for the (1+1)D transmission line problem in §4.3.9, and for the parallel-plate problem in §4.4.4. Boundary termination for vibrating beam and plate systems is discussed in detail in Chapter 5.