Comment: Passivity and Stability

At this point, we would like to mention an interesting property of the interleaved waveguide networks discussed in the earlier part of this section. We showed, in the last few pages, that three different types of immittance settings for the waveguide network could be used to solve the (1+1)D transmission line equations, and could, in fact, be interpreted as centered difference approximations. The three types of network integrate system (4.17) using slightly different effective inductances $\bar{l}$ and capacitances $\bar{c}$ which converge to $l$ and $c$ in the limit as the grid spacing becomes small. We had, for integer $i$,

$$\bar{l}_{i+\frac{1}{2}} = \frac{1}{2}\left(l_{i}+l_{i+1}\right) \bar{c}_{i} = c_{i}$$ Type I

$$\bar{l}_{i+\frac{1}{2}} = l_{i+\frac{1}{2}} \bar{c}_{i} = \frac{1}{2}\left(c_{i+\frac{1}{2}}+c_{i-\frac{1}{2}}\right)$$ Type II

$$\bar{l}_{i+\frac{1}{2}} = l_{i+\frac{1}{2}} \bar{c}_{i} = c_{i}$$ Type III

The three types, however, yield different requirements for passivity on the space step/time step ratio $v_{0}$,

$$v_{0} \geq \max_{i}\left(\sqrt{\frac{1}{l_{i}c_{i}}}\right)$$ Type I (4.51)

$$v_{0} \geq \max_{i+\frac{1}{2}}\left(\sqrt{\frac{1}{l_{i+\frac{1}{2}}c_{i+\frac{1}{2}}}}\right)$$ Type II (4.52)

$$v_{0} \geq \sqrt{\frac{1}{\min_{i+\frac{1}{2}}l_{i}\min_{i}c_{i}}}$$ Type III (4.53)

The first two bounds are roughly the same, and are close to optimal, in the sense that $v_{0}$ is bounded by(in the limit as $\Delta$ approaches 0) the maximum of the local group velocity over the transmission line. The type III bound, however, may be substantially poorer, and is similar to that which arises in MDWD networks (see §3.7).

If we choose $l(x)$ and $c(x)$ to be positive affine functions (linear in $x$ with a constant offset), then $\bar{l}$ and $\bar{c}$ are the same in all three cases, so the three networks will, in infinite-precision arithmetic, calculate identical solutions. But there will be a range of values of $v_{0}$ (namely, the range of $v_{0}$ greater than the bounds given in (4.40) and (4.41), but less than that of (4.42)) for which the type I and II networks are concretely passive [12], but for which the type III network is not. Over this range, some immittances in the type III network will necessarily be negative.

We can conclude that there is a large middle ground between passivity and global stability of networks. One important difference would seem to be that wave quantities in a concretely passive network are power-normalizable, whereas if a network is only abstractly passive--that is, its global behavior is passive, even though it contains elements which are themselves not--may not be. We do not investigate this further here, but comment that it would be of great interest to make clearer the distinction between passive and stable numerical methods for solving PDEs. This subject has been broached in some detail for ODEs [32,75], and we will see some other interesting examples of this distinction in Appendix A.