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Next: Comment: Passivity and Stability Up: Varying Coefficients Previous: Type II: Current-centered Network

Type III: Mixed Network

Suppose we set all the impedances which connect one grid point to another to be equal to some constant $ Z_{const}$ which is independent of position. Thus

$\displaystyle Y_{x^{-},i} = Y_{x^{+},i} = 1/Z_{const}$    

We then choose, to satisfy (4.33) and (4.34),
$\displaystyle Y_{c,i}$ $\displaystyle =$ $\displaystyle 2v_{0}c_{i}-\frac{2}{Z_{const}}$  
$\displaystyle Z_{c,i+\frac{1}{2}}$ $\displaystyle =$ $\displaystyle 2v_{0}l_{i+\frac{1}{2}}-2Z_{const}$  

and this leads to the conditions

$\displaystyle v_{0}\geq \max_{i}\frac{1}{c_{i}Z_{const}}=\frac{1}{c_{min}Z_{con...
...v_{0}\geq \max_{i}\frac{Z_{const}}{l_{i+\frac{1}{2}}}=\frac{Z_{const}}{l_{min}}$    

where

$\displaystyle c_{min} = \min_{i}c_{i}\hspace{1.0in}l_{min} = \min_{i}l_{i+\frac{1}{2}}$    

The lower bounds on $ v_{0}$ coincide when

$\displaystyle Z_{const} = \sqrt{l_{min}/c_{min}}$    

in which case we have

$\displaystyle v_{0} \geq \sqrt{\frac{1}{l_{min}c_{min}}}$    

Since in general, $ l_{min}c_{min}\leq\min_{i}(l_{i}c_{i})$, for $ 2i$ either even or odd, we are no longer at the optimal bound, and are forced to use a smaller time step than in the previous two cases, if we wish the network to remain concretely passive. This arrangement bears a strong resemblance to the MDWD network in [107] and [131], and discussed in §3.7. We will explore this similarity in more detail in §4.10. Many other choices are of the waveguide immittances satisfying (4.33) and (4.34) are of course possible.
next up previous
Next: Comment: Passivity and Stability Up: Varying Coefficients Previous: Type II: Current-centered Network
Stefan Bilbao 2002-01-22