Type I: Voltage-centered Network

We set the admittances of the waveguides leading away from a parallel junction to be identical, i.e.,

$$Y_{x^{-},i} = Y_{x^{+},i} = v_{0}c_{i}$$ (4.46)

and set

$$Y_{c,i,j} = 0$$

which satisfies (4.33) with $\bar{c}_{i} = c_{i}$. From (4.35), we have that

$$Z_{x^{+},i+\frac{1}{2}} = \frac{1}{v_{0}c_{i+1}}\hspace{1.0in}Z_{x^{-},i+\frac{1}{2}} = \frac{1}{v_{0}c_{i}}$$

Thus the series junction impedance at location $i+\frac{1}{2}$ will be

$$Z_{J,i+\frac{1}{2}} = \frac{1}{v_{0}c_{i}}+\frac{1}{v_{0}c_{i+1}}+Z_{c,i+\frac{1}{2}}$$

We can then set

$$Z_{c,i+\frac{1}{2}} = v_{0}(l_{i}+l_{i+1})-\frac{1}{v_{0}}\left(\frac{1}{c_{i}}+\frac{1}{c_{i+1}}\right)$$

which satisfies (4.34) with $\bar{l}_{i+\frac{1}{2}} = \frac{1}{2}(l_{i}+l_{i+1})$.

Only the series self-loop impedances are possibly negative, so the network will be passive if $Z_{c,i+\frac{1}{2}}\geq 0$. This will certainly be true if we choose

$$v_{0}\geq \max_{i}\left(\sqrt{\frac{1}{l_{i}c_{i}}}\right)$$ (4.47)

Recall that in our earlier discussion of group velocities for symmetric hyperbolic systems in §3.2 and for the transmission line in particular in §3.7, the maximum group velocity for the transmission line is

$$\gamma_{TL,max}^{g} = \max_{x\in\mathcal{D}}\frac{1}{\sqrt{lc}}$$ (4.48)

The optimal space step/time step ratio from (4.36) is exactly the maximum of the local group velocity of the transmission line, at least over the range of values of $l$ and $c$ sampled at the parallel junction locations; thus it approaches the maximum group velocity for the continuous system in the limit as the grid spacing $\Delta$ becomes small.