Initial Conditions

We will examine here the initialization of the MDWD network for the source-free transmission line system (3.56) with . This system requires initial distributions for both the current and voltage, which we will call and , respectively. For the initialization of the MDWD network for this system (shown in Figure 3.22), we will also need their spatial derivatives (assuming they exist), which we will write as and . We note that in the approach considered in [106], spatial derivative information has not been taken into account.

For the MDWD network of Figure 3.22, we must initialize all the wave variables incident upon the scattering junctions, written as , ; because we have assumed no sources, no wave enters through the loss/source port. This circuit is a MD representation, and each of these wave variables refers to an array. Assuming that the spatial grid spacing is , and the time step is , we can index the elements of these arrays as , for and integer; this represents an instance of the MD wave variable at grid location , and at time . For initialization, we must thus set , over all grid locations included in the domain of the problem, in terms of the quantities , and their spatial derivatives.We will consider only the settings for the wave variables in the left-hand adaptor; one proceeds in the same way for the right adaptor. We recall that the port resistances are defined by

Since the port resistances and are functions of position, we will write and . is independent of . It is easy to see, from (2.30), that the initial values and must be arranged such that we produce the initial current . Thus we need

Another condition is required to fully specify the wave variable initial values. Referring to the generating MDKC for this MDWD network in Figure 3.14(a), we can see that the voltage across the inductor of inductance will be . We intend to relate this voltage to the associated digital voltage across the inductor of port resistance in Figure 3.22. We have

At time , and at location , we may write this voltage as

The wave digital voltage across the same port, at location is defined by

Thus, for initialization, equating the voltages in (3.88) and (3.87), we must have

This requirement, along with (3.86) fully specifies the initial values of the wave variables at the left adaptor. We thus have

We note that may be obtained from the initial voltage distribution by any reasonable (i.e., consistent) approximation to the spatial derivative.

It is important to recognize that for constant , we have

These values occur in the same ratio as those of an

This rule is to be interpreted in a distributed sense, i.e., it holds for every instance of an adaptor on the numerical grid. A similar rule holds for a parallel adaptor. These settings ignore spatial derivative information, but give a simple way of proceeding in general, especially during the first stages of programming and debugging, and are correct (to first order) in the limit as approaches 0. If losses are large, though, one may prefer to use exact conditions like (3.89). This rule applies regardless of the number of dimensions of the problem (but may need to be amended if sources or reflection-free ports are present).