If we choose and to be positive affine functions (linear in with a constant offset), then and 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 (namely, the range of 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 , 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.