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Type I: Voltage-Centered Mesh

$\displaystyle Y_{\rho^{+},i,j} = Y_{\rho^{-},i,j} = Y_{\theta^{+},i,j} = Y_{\theta^{-},i,j} = \frac{\Delta_{\rho}c_{u,i,j}}{2T}\hspace{0.3in}Y_{c,i,j} = 0$ (4.97)

$\displaystyle Z_{c,i+\frac{1}{2},j}$ $\displaystyle =$ $\displaystyle \frac{\Delta_{\rho}l_{\rho,i,j}}{T}-\frac{2T}{\Delta_{\rho}c_{u,i,j}} + \frac{\Delta_{\rho}l_{\rho,i+1,j}}{T}-\frac{2T}{\Delta_{\rho}c_{u,i+1,j}}$ (4.98)
$\displaystyle Z_{c,i,j+\frac{1}{2}}$ $\displaystyle =$ $\displaystyle \frac{\Delta_{\theta}l_{\theta,i,j}}{T}-\frac{2T}{\Delta_{\rho}c_...
...+ \frac{\Delta_{\theta}l_{\theta,i,j+1}}{T}-\frac{2T}{\Delta_{\rho}c_{u,i,j+1}}$ (4.99)

The stability constraints (which follow from the requirement of positivity of $ Z_{c}$ everywhere) are

$\displaystyle \frac{\Delta_{\rho}}{T}\geq \max_{i,j}\sqrt{\frac{2}{l_{i,j}c_{i,...
...{T}\geq\max_{i,j}\left(\frac{1}{\rho_{i}}\sqrt{\frac{2}{l_{i,j}c_{i,j}}}\right)$ (4.100)

There is thus a dependence on $ \rho$ in the second condition (relating the angular spacing $ \Delta_{\theta}$ to $ T$), which we expect, since the spacing between the junctions at a given radius now varies linearly with the radius. Stability bounds are, for a radial mesh, necessarily more severe than in the rectilinear case, due to this variation in grid spacing.

Stefan Bilbao 2002-01-22