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Grid Arrangement Requiring Voltage and Tangential Current Density Component on Boundary

Consider the termination arrangement of Figure 4.23(a). In the source-free case, if for $ y=0$ we have $ u(x,0,t)=0$, then from (4.58a),

$\displaystyle l\frac{\partial i_{x}}{\partial t} + ri_{x} = 0$   for$\displaystyle \hspace{0.3in} y=0$    

Thus the current density component tangential to the boundary is uncoupled from the other dependent variables. It is convenient to assume, then, that $ i_{x}(x,0,t)$ is initially zero, so that it will remain so permanently. In this case, we can drop the series junctions corresponding to $ i_{x}$ on the southern boundary from the network. Otherwise, we may allow the junctions to remain, as lumped damped (by a factor $ r/l$) elements, still uncoupled from the rest of the network. In either case, the parallel junctions at the grey points in Figure 4.23(a) may be short-circuited as in the (1+1)D case in order to realize boundary condition (4.68). The waveguide mesh termination corresponding to (4.68) is shown in Figure 4.24(a).

To deal with the boundary condition $ i_{y}(x,0,t) = 0$, we may proceed as in the (1+1)D case, and write down a difference approximation to (4.58c), where we use the one-sided difference approximation

$\displaystyle \frac{\partial i_{y}}{\partial y}\Big\vert _{y=0}$ $\displaystyle =$ $\displaystyle \frac{2}{\Delta}\left(i_{y}(x,\frac{\Delta}{2},t)-i_{y}(x,0,t)\right)+O(\Delta)\notag$  
  $\displaystyle =$ $\displaystyle \frac{2}{\Delta}i_{y}(x,\frac{\Delta}{2},t)+O(\Delta)$  

and centered differences in the time and $ x$ directions,
$\displaystyle U_{i,0}(n) - \rho_{U,i,0}U_{i,0}(n-1)$ $\displaystyle +$ $\displaystyle \sigma_{U,i,0}\left(I_{x,i+\frac{1}{2},0}(n-{\textstyle \frac{1}{2}})-I_{x,i-\frac{1}{2},0}(n-{\textstyle \frac{1}{2}})\right)\notag$  
  $\displaystyle +$ $\displaystyle 2\sigma_{U,i,0}I_{y,i,\frac{1}{2}}(n-{\textstyle \frac{1}{2}})= 0$  

where $ \sigma_{U,i,0}$ and $ \rho_{U,i,0}$ are as given in (4.60).

Here, the voltages on the boundary are related to the tangential currents, which, from Figure 4.23(a), are also calculated on the boundary. This implies that the corresponding junctions will be connected to one another by waveguides which lie directly on the boundary. Also notice the doubled weighting of the $ I_{y}$ grid function at the boundary; this requires special care in the DWN implementation, though it also follows from a structurally passive termination, provided we make use of transformers along the boundary waveguides. Though we have not discussed transformers in the DWN context [166], they are identical to wave digital transformers, which were mentioned in §2.3.4. In effect, we may introduce multiplies by $ \kappa$ and $ 1/\kappa$, for any real $ \kappa$ in the two signal paths of any waveguide without affecting losslessness, provide we scale impedances at both ends of the waveguide accordingly. The DWN termination corresponding to $ i_{y} = 0$ at a southern boundary is shown in Figure 4.24(b). We have used transformers of turns ratio 2, implying that the immittances on the boundary satisfy

$\displaystyle Y_{x^{-},i,0} = \frac{1}{4Z_{x^{+},i-\frac{1}{2},0}}\hspace{1.0in}Y_{x^{+},i,0} = \frac{1}{4Z_{x^{-},i+\frac{1}{2},0}}$ (4.82)

The corresponding difference equation at a parallel boundary junction is then
$\displaystyle U_{J,i,0}(n) - \frac{Y_{J,i,0}-2Y_{R,i,0}}{Y_{J,i,0}}U_{J,i,0}(n-1)$ $\displaystyle +$ $\displaystyle \frac{1}{Y_{J,i,0}}\left(I_{xJ,i+\frac{1}{2},0}(n-{\textstyle \frac{1}{2}})-I_{xJ,i-\frac{1}{2},0}(n-{\textstyle \frac{1}{2}})\right)\notag$  
  $\displaystyle +$ $\displaystyle \frac{2}{Y_{J,i,0}}I_{y,i,\frac{1}{2}}(n-{\textstyle \frac{1}{2}})= 0$  


$\displaystyle Y_{J,i,0}\triangleq Y_{x^{-},i,0}+Y_{x^{+},i,0}+Y_{y^{+},i,0}+Y_{c,i,0}+Y_{R,i,0}$ (4.83)

The junction updating will be equavlent to the centered difference scheme if we choose

$\displaystyle Y_{J,i,0} = v_{0}\bar{c}_{i,0}+\frac{g_{i,0}\Delta}{2}\hspace{1.0in}Y_{R,i,0} = \frac{g_{i,0}\Delta}{2}$ (4.84)

Given that the northward immittances at the boundary junctions must be set as interior values, the settings for the remaining immittances for the type I and II meshes discussed in §4.4.2 will be

Type I:

$\displaystyle Y_{x^{-},i,0}$ $\displaystyle =$ $\displaystyle Y_{x^{+},i,0} = \frac{v_{0}c_{i,0}}{4}\hspace{0.5in}Z_{x^{+},i+\f...
...{0}c_{i+1,0}}\hspace{0.5in}Z_{x^{-},i+\frac{1}{2},0} = \frac{1}{v_{0}c_{i-1,0}}$  
$\displaystyle Y_{c,i,0}$ $\displaystyle =$ $\displaystyle 0\hspace{0.5in}Z_{c,i+\frac{1}{2},0} = v_{0}l_{i+1,0}+v_{0}l_{i,0}-\frac{1}{v_{0}c_{i+1,0}}-\frac{1}{v_{0}c_{i,0}}$  

Type II:

$\displaystyle Z_{x^{-},i+\frac{1}{2},0}$ $\displaystyle =$ $\displaystyle Z_{x^{-},i+\frac{1}{2},0} = v_{0}l_{i+\frac{1}{2},0}\hspace{0.5in...
...frac{1}{2},0}}\hspace{0.5in}Y_{x^{-},i,0} = \frac{1}{4v_{0}l_{i-\frac{1}{2},0}}$  
$\displaystyle Z_{c,i+\frac{1}{2},0}$ $\displaystyle =$ $\displaystyle 0\hspace{0.5in}Y_{c,i,0} = \frac{v_{0}c_{i,\frac{1}{2}}}{2}+ \fra...

The passivity conditions which follow from the positivity of the boundary self-loop immittances will be
$\displaystyle v_{0}$ $\displaystyle \geq$ $\displaystyle \max_{i}\sqrt{\frac{1}{l_{i,0}c_{i,0}}}$   Type I  
$\displaystyle v_{0}$ $\displaystyle \geq$ $\displaystyle \max\left(\max_{i}\sqrt{\frac{2}{l_{i,\frac{1}{2}}c_{i,\frac{1}{2...
...x_{i+\frac{1}{2}}\sqrt{\frac{1}{l_{i+\frac{1}{2},0}c_{i+\frac{1}{2},0}}}\right)$   Type II  

and clearly are less restrictive than the conditions (4.63) and (4.67) over the mesh interior, and hence do not affect the overall stability bound on $ v_{0}$.

Figure 4.24: (2+1)D waveguide mesh terminations at a southern boundary, for the grid arrangement of Figure 4.23(a)-- (a) for $ u(x,0,t)=0$ and (b) $ i(x,0,t) = 0$.
...\tiny {$U_{y^{+}}^{+}$}}
\end{picture} \vspace{0.3in}
\end{center} \end{figure}

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Next: Grid Arrangement with Normal Up: Boundary Conditions Previous: Boundary Conditions
Stefan Bilbao 2002-01-22