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Digression: Derivation of an Inductive Lattice Twoport
We have derived the WD equivalents for all the standard circuit elements, but the twoports pictured in Figure 3.12 need a special treatment. Fettweis and Nitsche find the lattice form to be the most straightforward derivation, but we would especially like to call attention to the fact that the resultant WD twoport is the same regardless of which of the equivalent structures we choose; use of a concretely MDpassive twoport, however, makes the passivity of the resulting circuit obvious. We will continue to use the Jaumann equivalent in all future diagrams (though we could equally well use the lattice form). Basu [10], as well as Fettweis [46] make the point that an electrical network equivalent is a convenient formalism for developing MDpassive discrete networks, but it is by no means necessary.
Beginning from any of the equivalent twoport structures in Figure 3.12, we can immediately write down the impedance matrix, which we will denote by
:
where we have set
and =
.
We now introduce a port resistance matrix
, where we can choose equal port resistances because the twoport is symmetric. The scattering matrix is then, from (2.21),
where
is the identity matrix. This is easily rearranged to become
Returning to the problem of the (1+1)D transmission line, for the twoport in question in Figure 3.14(a), we have
and
. Notice in particular that for these choices of impedance, any of the twoports of Figure 3.12 are described by MD positive real matrices. We now discretize using the trapezoid rule, i.e.
where we also will set
. If we make the choice
, we obtain the discretetime, causal scattering matrix

(3.58) 
The resulting MD twoport is shown in Figure 3.13.
Figure 3.13:
Signal flow diagram for the MDWD lattice or Jaumann twoport.

It should be clear that the same procedure can be used for arbitrary impedances and ; it should be remarked, however, that we can not always get a simple form without a through path like that pictured in Figure 3.13. Since this twoport is strictly causal, it may easily be connected to other ports without the risk of the appearance of a delayfree loop.
Next: A MDWD Network for
Up: The (1+1)D Transmission Line
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Stefan Bilbao
20020122