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Alternative MDKC for the (1+1)D Transmission Line

We now reexamine the lossless, source-free (1+1)D transmission line equations and show that how, after a simple network manipulation and under the alternative spectral mappings mentioned in §3.5.4, we end up with an interleaved DWN identical to that discussed in §4.3.6.

We begin by first scaling the (1+1)D transmission line equations by a factor of $ \Delta$, and where, as before, we introduce a scaled time variable defined by $ t' = v_{0}t$, giving

$\displaystyle \begin{eqnarray}\Delta v_{0}l\frac{\partial i}{\partial t'} + \De...
...tial i}{\partial t'} + \Delta\frac{\partial u}{\partial x} &=& 0 \end{eqnarray}$ (4.123a)

It should be clear that the scaling by $ \Delta$ will have no effect on the solution to equations (4.106), even in the limit as $ \Delta\rightarrow 0$. The MDKC for this system is shown in Figure 4.47(a), where we have used the coordinate transformation defined by (3.18). Aside from the scaling of the element values by $ \Delta$, this is identical to the MDKC of Figure 3.14(a), where the resistors and voltage sources (corresponding to loss and source terms) have been omitted (they can be simply reintroduced at a later stage).

Figure 4.47: MDKCs for the lossless source-free (1+1)D transmission line equations-- (a) the standard representation and (b) a modified form.
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...(a)}
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\end{figure}

The element values are given by

$\displaystyle L_{1} = \Delta\left(v_{0}l-r_{0}\right)\hspace{0.3in}L_{2} = \Delta\left(v_{0}cr_{0}^{2}-r_{0}\right)\hspace{0.3in}L_{0} = \Delta r_{0}/\sqrt{2}$    

In this representation, the transmission line voltage $ u$ is considered, after a scaling by $ 1/r_{0}$, to be a current. This is somewhat unsatisfying from a physical point of view; it is easy, however, to rectify this: by a simple network transformation, the inductor in the right-hand loop (through which current $ u/r_{0}$ flows) may be replaced by a gyrator of gyration constant $ r_{0}$ terminated on a capacitor. The voltage across this capacitor of capacitance $ C_{2} = \Delta\left(v_{0}c-\frac{1}{r_{0}}\right)$ will then be exactly $ u$. See Figure 4.47(b).

Consider now the two-port of Figure 4.47(b), with terminals $ A$, $ A'$, $ B$ and $ B'$. The hybrid matrix for this linear shift-invariant two-port is, in terms of the frequencies $ s_{1}$ and $ s_{2}$,

$\displaystyle {\bf K}(s_{1},s_{2}) = \frac{\Delta}{\sqrt{2}}\begin{bmatrix}r_{0...
...{1}-s_{2}\\ s_{1}-s_{2}&\frac{1}{r_{0}}\left(s_{1}+s_{2}\right)\\ \end{bmatrix}$    

We now apply the alternative spectral mapping introduced in §3.5.4, namely

$\displaystyle s_{1}\rightarrow \frac{1}{T_{1}}\frac{(1-z_{1}^{-1})(1+z_{2}^{-1}...
...rrow \frac{1}{T_{2}}\frac{(1-z_{2}^{-1})(1+z_{1}^{-1})}{1+z_{1}^{-1}z_{2}^{-1}}$ (4.124)

with $ T_{1} = T_{2} = \Delta/\sqrt{2}$ (recall that the for the interleaved scheme, we will have increments in the time step of $ T/2$ and in the space step of $ \Delta/2$, and thus we have chosen $ T_{1}$ and $ T_{2}$ to be half the values they previously took). It will be recalled that this mapping, like the trapezoidal rule, is passivity-preserving. The hybrid matrix becomes

$\displaystyle {\bf K}(z_{1}^{-1},z_{2}^{-1}) = \frac{2}{1+z_{1}^{-1}z_{2}^{-1}}...
...}-z_{1}^{-1}&\frac{1}{r_{0}}\left(1-z_{1}^{-1}z_{2}^{-1}\right)\\ \end{bmatrix}$ (4.125)

which, upon inspection, can be written as the sum

$\displaystyle {\bf K}(z_{1}^{-1},z_{2}^{-1}) = {\bf K}_{ue}(z_{1}^{-1},z_{2}^{-1})+{\bf K}_{ue}(-z_{2}^{-1},-z_{1}^{-1})$    

where it will be recalled that $ {\bf K}_{ue}(z_{1}^{-1},z_{2}^{-1})$ is the hybrid matrix for the multidimensional unit element defined in (4.104), with $ R = r_{0}$. Because for a series/parallel connection of two-ports, hybrid matrices sum, we have thus decomposed our connecting two-port into two multidimensional unit elements of opposing directions, one of which incorporates a sign-inversion in both of its signal paths. The MDWD network can then immediately be constructed as in Figure 4.48. The port resistances are

$\displaystyle R_{1} = \frac{2}{\Delta}L_{1} = 2v_{0}l-2r_{0}\hspace{0.5in}R_{2} = \frac{\Delta}{2C_{2}} = \frac{1}{2v_{0}c-2/r_{0}}\hspace{0.5in}R_{0} = r_{0}$    

We have chosen here a doubled delay length (of $ T' = \Delta$) in the self-loops, according to the offset scheme mentioned in §3.9 (for these one-ports, we use the trapezoid rule as for MDWD networks).

Figure 4.48: MD network equivalent to type III DWN for the lossless source-free (1+1)D transmission line equations under an alternative spectral mapping.
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...ut(309,66){\footnotesize {${\bf T}$}}
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This network is, when spatial dependence is expanded out, identical to the DWN shown in Figure 4.44, under the replacement of the series and parallel adaptor symbols by series and parallel scattering junction symbols--recall that they perform identical operations. This MDWD network, then, (if it can be called that) operates on a decimated grid, unlike the standard form shown in Figure 4.44. Losses and sources can easily be added back in to the alternative MDKC of Figure 4.47(b), and the resulting MDWD network will be identical to that of Figure 4.15.

We would conjecture that it is possible to find similar equivalences for the type I and II DWNs for the same system (which have better stability bounds); in these cases however, recall from §4.3.6 that the immittances of the connecting waveguides did indeed vary from one grid location to the next. In order to design a MDKC corresponding to such a network, it would be necessary to extend the definition of the multidimensional unit element to include spatially-varying port-resistances (indeed, this extension is automatic, since the port resistances do not appear explicitly in the definition of this element in (4.104)). The problem, then, is that the two-port connecting the series and parallel adaptors will no longer be shift-invariant, so we must take special care with application of the discretization rule, which can no longer be treated as a spectral mapping.


next up previous
Next: Alternative MDKC for (2+1)D Up: Incorporating the DWN into Previous: Hybrid Form of the
Stefan Bilbao 2002-01-22