A MDWD Network for the (1+1)D Transmission Line

Returning to Figure 3.14(a), and making use of the discrete two-port derived in the last section, we can now write the complete wave digital network. It is shown in Figure 3.14(b).

Figure 3.14: (a) MD-passive network for the (1+1)D transmission line equations and (b) its associated MDWD network.

The inductances in the MDKC of Figure 3.14(a) are

$$L_{1} = v_{0}l-r_{0}\hspace{0.5in}L_{2} = v_{0}r_{0}^{2}c-r_{0}\hspace{0.5in}L_{0} = \frac{r_{0}}{\sqrt{2}}$$ (3.59)

and the port resistances of the MDWD network of Figure 3.14(b) are

$$R_{0} = \frac{2r_{0}}{\Delta}\hspace{0.3in}R_{1} = \frac{2L_{1}}{... ... \frac{2L_{2}}{\Delta}\hspace{0.3in}R_{gh} = gr_{0}^{2}\hspace{0.3in}R_{er} = r$$ (3.60)

The MDWD network is MD-passive if all the port resistances are non-negative over the entire spatial domain; from (3.63) and (3.64), the only port resistances which are possibly negative are $R_{1}$ and $R_{2}$. Requiring their positivity gives the constraints

$$v_{0}\geq \frac{r_{0}}{l_{min}}\hspace{1.0in}v_{0}\geq\frac{1}{r_{0}c_{min}}$$

where $l_{min} = \min_{x}l$ and $c_{min} = \min_{x}c$. A judicious choice of $r_{0} = \sqrt{\frac{l_{min}}{c_{min}}}$ [131] allows the largest possible time step for a given grid spacing; the condition is then

$$v_{0} \geq \sqrt{\frac{1}{l_{min}c_{min}}}\geq \gamma_{TL,max}^{g}$$ (3.61)

where $\gamma_{TL,max}$ is defined by (3.59).

If $l$ and $c$ are constant, and (3.65) holds with equality, so that we have

$$v_{0} = \sqrt{\frac{1}{lc}} = \gamma_{TL,max}^{g}$$ (3.62)

then the MDWD numerical scheme is said to be operating at the Courant-Friedrichs-Lewy (CFL) bound [176]. For varying coefficients, however, $v_{0}$ is bounded away from $\gamma_{TL,max}^{g}$, so the time step will have to be chosen smaller than might be expected; we will look at how to improve upon this bound in §3.12.