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Next: Losses, Sources, and Spatially-varying Up: The (2+1)D Parallel-plate System Previous: Defining Equations and Centered

The Waveguide Mesh

Consider the original form (2+1)D waveguide network, or mesh [198], operating on a rectilinear grid. Each scattering junction (parallel) is connected to four neighbors by unit sample bidirectional delay lines. The spacing of the junctions is $ \Delta$ (in either the $ x$ or $ y$ direction) and the time delay is $ T$ in the delay lines (see Figure 4.19). We now index a junction (and all its associated voltages and currents and wave quantities) at coordinates $ (i\Delta,j\Delta)$ by the pair $ (i,j)$.

Figure 4.19: (2+1)D waveguide mesh and a representative scattering junction.
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As in the (1+1)D case, at each parallel junction at location $ (i,j)$, we have voltages at every port, given by
$\displaystyle U_{x^{+},i,j}$ $\displaystyle =$ voltage in waveguide leading east$\displaystyle \notag$  
$\displaystyle U_{x^{-},i,j}$ $\displaystyle =$ voltage in waveguide leading west$\displaystyle \notag$  
$\displaystyle U_{y^{+},i,j}$ $\displaystyle =$ voltage in waveguide leading north$\displaystyle \notag$  
$\displaystyle U_{y^{-},i,j}$ $\displaystyle =$ voltage in waveguide leading south$\displaystyle \notag$  

and current flows
$\displaystyle I_{x^{+},i,j}$ $\displaystyle =$ current flow in waveguide leading east$\displaystyle \notag$  
$\displaystyle I_{x^{-},i,j}$ $\displaystyle =$ current flow in waveguide leading west$\displaystyle \notag$  
$\displaystyle I_{y^{+},i,j}$ $\displaystyle =$ current flow in waveguide leading north$\displaystyle \notag$  
$\displaystyle I_{y^{-},i,j}$ $\displaystyle =$ current flow in waveguide leading south$\displaystyle \notag$  

as well as wave quantities
$\displaystyle U_{q,i,j}$ $\displaystyle =$ $\displaystyle U_{q,i,j}^{+}+U_{q,i,j}^{-}$  
$\displaystyle I_{q,i,j}$ $\displaystyle =$ $\displaystyle I_{q,i,j}^{+}+I_{q,i,j}^{-}$  

where $ q$ is any of $ x^{+}$, $ x^{-}$, $ y^{+}$ or $ y^{-}$. The variables superscripted with a $ +$ refer to the incoming waves, and those marked $ -$ to outgoing waves. The voltage and current waves are related by

$\displaystyle I_{q,i,j}^{+} = Y_{q,i,j}U_{q,i,j}^{+}\hspace{1.5in}I_{q,i,j}^{-} = -Y_{q,i,j}U_{q,i,j}^{-}$ (4.67)

where $ Y_{q,i,j}$ is the admittance of the waveguide connected to the junction with coordinates $ (i\Delta,j\Delta)$ in direction $ q$. The junction admittance is then

$\displaystyle Y_{J,i,j} \triangleq Y_{x^{-},i,j}+Y_{x^{+},i,j}+Y_{y^{-},i,j}+Y_{y^{+},i,j}$ (4.68)

and the scattering equation, for voltage waves, will be, from (4.15),

$\displaystyle U_{r,i,j}^{-} = -U_{r,i,j}^{+} + \frac{2}{Y_{J,i,j}}\left(Y_{x^{-...
...},i,j}^{+}+Y_{y^{+},i,j}U_{y^{+},i,j}^{+}+Y_{y^{-},i,j}U_{y^{-},i,j}^{+}\right)$ (4.69)

where $ r$ is any of $ x^{+}$, $ x^{-}$, $ y^{+}$, or $ y^{-}$. Voltage waves are propagated by:

$\displaystyle U_{x^{+},i,j}^{+}(n)$ $\displaystyle = U_{x^{-},i+1,j}^{-}(n-1)$   $\displaystyle U_{x^{-},i,j}^{+}(n) = U_{x^{+},i-1,j}^{-}(n-1)$    
$\displaystyle U_{y^{+},i,j}^{+}(n)$ $\displaystyle = U_{y^{-},i,j+1}^{-}(n-1)$   $\displaystyle U_{y^{-},i,j}^{+}(n) = U_{y^{+},i,j-1}^{-}(n-1)$    

The case of flow waves is similar except for a sign inversion. The complete picture is shown in Figure 4.19.

Similarly to the (1+1)D case, it is possible to obtain a finite difference scheme purely in terms of the junction voltages $ U_{J,i,j}$, under the assumption that the admittances of all the waveguides in the network are identical, and equal to some positive constant $ Y$. Thus, from (4.56), $ Y_{J,i,j} = 4Y$. We have, for the junction at location $ x=i\Delta$, $ y=j\Delta$,

$\displaystyle U_{J,i,j}(n+1)\hspace{0.1in}$ $\displaystyle =\hspace{0.1in}$   $\displaystyle \frac{2}{Y_{J,i,j}}\sum_{q}Y_{q,i,j}U_{q,i,j}^{+}(n+1),\hspace{0.5in}q=\{x^{-},x^{+},y^{-},y^{+}\}\notag$    
  $\displaystyle =$   $\displaystyle \frac{1}{2}\left(U_{x^{-},i+1,j}^{-}(n)+U_{x^{+},i-1,j}^{-}(n)+U_{y^{-},i,j+1}^{-}(n) + U_{y^{+},i,j-1}^{-}(n)\right)\notag$    
  $\displaystyle =$   $\displaystyle \frac{1}{2}\left( U_{J,i+1,j}(n)+U_{J,i-1,j}(n)+U_{J,i,j+1}(n) + U_{J,i,j-1}(n)\right)\notag$    
      $\displaystyle -\hspace{0.1in}\frac{1}{2}\left( U_{x^{-},i+1,j}^{+}(n)+U_{x^{+},i-1,j}^{+}(n)+U_{y^{-},i,j+1}^{+}(n) + U_{y^{+},i,j-1}^{+}(n)\right)\notag$    
  $\displaystyle =$   $\displaystyle \frac{1}{2}\left( U_{J,i+1,j}(n)+U_{J,i-1,j}(n)+U_{J,i,j+1}(n) + U_{J,i,j-1}(n)\right)\notag$    
      $\displaystyle -\hspace{0.1in}\frac{2}{Y_{J,i,j}}\sum_{q}Y_{q,i,j}U_{q,i,j}^{-}(n-1)\notag$    
  $\displaystyle =$   $\displaystyle \frac{1}{2}\left( U_{J,i+1,j}(n)+U_{J,i-1,j}(n)+U_{J,i,j+1}(n) + U_{J,i,j-1}(n)\right)-U_{J,i,j}(n-1)$    

This is identical to % latex2html id marker 83627
$ \ref{magicdiff2d}$ if we replace $ U_{J}$ by $ U$.

If we now replace all the bidirectional delay lines in Figure 4.19 by the same split pair of lines shown in Figure 4.11, then we get the arrangement in Figure 4.20.

Figure 4.20: (2+1)D interleaved waveguide mesh.
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We have placed the split lines such that the branches containing sign inversions are adjacent to the western and southern ports of the parallel junctions. We also introduce new junction variables $ I_{xJ}$ at the series junctions between two horizontal half-sample waveguides, and $ I_{yJ}$ at the series junctions between two vertical delay lines, as well as all the associated wave quantities at the ports of the new series junctions. It is straightforward to show that upon identifying $ I_{xJ}$, $ I_{yJ}$ and $ U_{J}$ with $ I_{x}$ and $ I_{y}$ and $ U_{J}$, the mesh will calculating according to scheme (4.52) with constant coefficients, if we choose

$\displaystyle v_{0} = \sqrt{\frac{2}{lc}}\hspace{1.0in}Z = \sqrt{\frac{2l}{c}}$    

where $ Z$ is the impedance in all the delay lines. We are again at the magic time step, but the impedance has been set to be larger than the characteristic impedance of the medium. Also, notice that the speed of propagation along the delay lines is not the wave speed of the medium, which is $ \gamma = 1/\sqrt{lc}$. Such a mesh is called a slow-wave structure [90] in the TLM literature. At this point, it is useful to compare Figures 4.20 and 4.18.

next up previous
Next: Losses, Sources, and Spatially-varying Up: The (2+1)D Parallel-plate System Previous: Defining Equations and Centered
Stefan Bilbao 2002-01-22