Phase and Group Velocity

For the constant-coefficient, lossless and source-free case (i.e., $r=g=e=f=h=0$), the numerical dispersion relation, in terms of the frequency $\omega$ and wavenumber magnitude $\Vert$$\beta$$\Vert _{2} = \sqrt{\beta_{x}^{2}+\beta_{y}^{2}}$ will be, from (3.10),

$$\omega\left(\omega^{2}-\frac{1}{lc}\Vert\mbox{\boldmath$\beta$}\Vert _{2}^{2}\right) = 0$$

which has roots

$$\omega = 0\hspace{1.0in}\omega = \pm\sqrt{\frac{1}{lc}}\Vert \mbox{\boldmath$\beta$} \Vert _{2}$$

Discounting the stationary mode with $\omega = 0$, the phase and group velocities are then, from (3.12),

$$\gamma^{p}_{PP} = \gamma_{PP}^{g} = \pm\frac{1}{\sqrt{lc}}$$

and if $l$ and $c$ are functions of $x$ and $y$, the maximal group velocity will be

$$\gamma_{PP, max}^{g} = \frac{1}{\sqrt{(lc)_{min}}}$$ (3.64)

where $(lc)_{min} = \min_{(x,y)\in\mathcal{D}}(lc)$. This bound is the same as for the (1+1)D transmission line equations.

Figure 3.17: MDKC for the (2+1)D parallel-plate system in rectangular coordinates.