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Central Gridpoint in a Radial Mesh

We have so far restricted our attention to interior points of the grid (for which $ i> 0$). If the center of the $ (\rho,\theta)$ coordinate system is to be contained in the grid, a special treatment is required. We have indexed the grid variables such that, in our interleaved mesh, a single parallel junction lies at the origin (for $ i=0$). If the problem domain includes a full circle, we also must assume that $ \Delta_{\theta}$ divides 2$ \pi$ evenly so that we have a positive integer $ N$ such that

$\displaystyle N = \frac{2\pi}{\Delta_{\theta}}$    

Thus the central parallel junction will be connected to $ N$ series junctions at locations $ (\frac{\Delta_{\rho}}{2},j\Delta_{\theta})$, $ j=0,\hdots, N-1$. We will name the admittances of the $ N$ waveguides radiating from the central hub $ Y_{j,0,0}$, $ j=0,\hdots, N-1$, and the admittance of the self-loop and loss/source ports will be $ Y_{c,0,0}$ and $ Y_{R,0,0}$ respectively (see Figure 4.31).

Figure 4.31: Central scattering junction for the waveguide mesh in radial coordinates.
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The difference scheme relating the junction voltage at the central grid point to the radial currents at the surrounding junctions will then be

$\displaystyle U_{J,0,0}(n)$ $\displaystyle -$ $\displaystyle \frac{Y_{J,0,0}-2Y_{R,0,0}}{Y_{J,0,0}}U_{J,0,0}(n-1) + \frac{2}{Y_{J,0,0}}\sum_{j=0}^{N-1}I_{\rho J,\frac{1}{2},j}(n-\frac{1}{2})\notag$ (4.105)
  $\displaystyle -$ $\displaystyle \frac{2Y_{R,0,0}}{Y_{J,0,0}}\left(U_{R,0,0}^{+}(n)+U_{R,0,0}^{+}(n-1)\right) = 0$ (4.106)

The sum over the junction currents, which we now look at in the continuous time/space domain so as to develop a series approximation) may be rewritten using (4.83) in terms of the rectilinear current variables as
$\displaystyle \sum_{j=0}^{N-1}i_{\rho}(\frac{\Delta_{\rho}}{2},j\Delta_{\theta})$ $\displaystyle =$ $\displaystyle \sum_{j=0}^{N-1}\frac{\Delta_{\rho}}{2}\left(i_{x}(\frac{\Delta_{...
...y}(\frac{\Delta_{\rho}}{2},j\Delta_{\theta})\sin(j\Delta_{\theta})\right)\notag$ (4.107)
  $\displaystyle =$ $\displaystyle \frac{\Delta_{\rho}}{2}i_{x}(0,0)\sum_{j=0}^{N-1}\cos(j\Delta_{\t...
...)+\frac{\Delta_{\rho}}{2}i_{y}(0,0)\sum_{j=0}^{N-1}\sin(j\Delta_{\theta})\notag$ (4.108)
    $\displaystyle \hspace{0.3in}+\hspace{0.1in} \frac{\Delta_{\rho}^{2}}{4}\sum_{j=...
...tial y}\right\vert _{(0,0)}\hspace{-0.1in}\sin^{2}(j\Delta_{\rho})\right)\notag$ (4.109)
  $\displaystyle =$ $\displaystyle \frac{N\Delta_{\rho}^{2}}{8}\left(\left.\frac{\partial i_{x}}{\pa...
...pace{-0.1in}+\left.\frac{\partial i_{y}}{\partial y}\right\vert _{(0,0)}\right)$ (4.110)

where we have neglected higher order terms in $ \Delta_{\rho}$ and used, in the last line, the identities

$\displaystyle \sum_{j=0}^{N-1}\cos\left(\frac{2\pi j}{N}\right) = \sum_{j=0}^{N...
...}{N}\right)=\sum_{j=0}^{N-1}\sin^{2}\left(\frac{2\pi j}{N}\right) = \frac{N}{2}$    

which hold for $ N>2$. Using approximation (4.94) and by comparing (4.93) with (4.82c), we must choose the junction and loss/source port admittance to be

$\displaystyle Y_{J,0,0} = \frac{1}{4}N\Delta_{\rho}^{2}\left(\frac{\bar{c}_{0,0...
...{0,0}}{2}\right)\hspace{0.5in} Y_{R,0,0} = \frac{1}{8}N\Delta_{\rho}^{2}g_{0,0}$    

and the source wave variable $ U_{R,0,0}^{+}(n)$ to be

$\displaystyle U_{R,0,0}^{+}(n) = -\frac{h_{0,0}(n)}{g_{0,0}}$    

A mesh of type I is infeasible because it would require access to $ l_{\rho,0,0}$ in order to set $ Z_{c,\frac{1}{2},j}$ as prescribed in (4.86), but $ l_{\rho}$ as defined in (4.84) is singular at the origin (although if we are working with a radial geometry which does not contain the origin, this problem does not arise). For a mesh of type II, we have, from (4.89),

$\displaystyle Y_{j,0,j} = \frac{1}{Z_{\rho^{-},\frac{1}{2},j}} = \frac{T}{\Delta_{\rho}l_{\rho,\frac{1}{2},j}} = \frac{T}{2l_{\frac{1}{2},j}}$    

and so we may set, for the self-loop admittance at the central junction,

$\displaystyle Y_{c,0,0} = \sum_{j=0}^{N-1}\left(\frac{\Delta_{\rho}^{2}c_{\frac{1}{2},j}}{4T} - \frac{T}{2l_{\frac{1}{2},j}}\right)$    

which is positive when

$\displaystyle \frac{\Delta_{\rho}}{T}\geq \max_{j=0,\hdots,N-1} \sqrt{\frac{2}{l_{\frac{1}{2},j}c_{\frac{1}{2},j}}}$    

Thus the stability requirement at the central junction does not interfere with the requirements over the interior of the mesh given for the type II mesh.

We note that a different type of central node, proposed for use in radial TLM simulations, is described in [24].

next up previous
Next: Simulation: Circular Region with Up: The Waveguide Mesh in Previous: Type II: Current-Centered Mesh
Stefan Bilbao 2002-01-22