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Defining Equations and Centered Differences

The set of PDEs describing a lossless, source-free parallel-plate transmission line in (2+1)D is a direct generalization of system (4.17):
$\displaystyle \begin{eqnarray}l\frac{\partial i_{x}}{\partial t} + \frac{\parti...
...tial i_{x}}{\partial x} + \frac{\partial i_{y}}{\partial y} &=& 0\end{eqnarray}$ (4.60a)

Now $ i_{x}(x,y,t)$ and $ i_{y}(x,y,t)$ are the components of the current density vector in the $ x$ and $ y$ directions, respectively, and $ u(x,y,t)$ is the voltage between the plates. $ l(x,y)$ and $ c(x,y)$, both assumed positive everywhere, are the inductance and capacitance per unit length.

If we assume that $ l$ and $ c$ are constant, then as in the (1+1)D case, the set of equations can be reduced to a single second order equation in the voltage alone:

$\displaystyle \frac{\partial^{2} u}{\partial t^{2}}=\gamma^{2}\Big(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}\Big)$ (4.61)

and again, the wave speed $ \gamma$ is given by

$\displaystyle \gamma = \frac{1}{\sqrt{lc}}$    

The centered difference scheme for system (4.49) also generalizes simply. Define grid functions $ I_{x.i,j}(n)$, $ I_{y,i,j}(n)$ and $ U_{i,j}(n)$ which run over half-integer values of $ i$, $ j$, and $ n$, i.e.,

$\displaystyle i,j,n = \hdots -1,-{\textstyle \frac{1}{2}},0,{\textstyle \frac{1}{2}},1\hdots$    

We will furthermore assume that the spatial step in the $ x$ direction and the $ y$ direction are the same and equal to $ \Delta/2$. As before, the time step will be $ T/2$. We can use the approximations (4.19), as well as an approximation to the derivative in the $ y$ direction,
$\displaystyle \left.\frac{\partial w}{\partial y}\right\vert _{i\Delta,j\Delta,nT}$ $\displaystyle =$ $\displaystyle \frac{w(i,j+\frac{1}{2},n)-w(i,j-\frac{1}{2},n)}{\Delta}+O(\Delta^{2})$  

where $ w$ stands for either of $ i_{y}$ or $ u$. We obtain the difference scheme
\begin{subequations}\begin{alignat}{2} \hspace{-0.5in}I_{x,i,j}(n+{\textstyle \f...
...{1}{2}}(n)-I_{y,i,j-\frac{1}{2}}(n)\right) &&= 0 \end{alignat}\end{subequations}

where we have written
$\displaystyle \bar{l}_{i,j} \triangleq l(i\Delta,j\Delta)+O(\Delta^{2})$      
$\displaystyle \bar{c}_{i,j} \triangleq l(i\Delta,j\Delta)+O(\Delta^{2})$      


$\displaystyle v_{0} \triangleq \frac{\Delta}{T}$    

As in the (1+1)D case, it is possible to subdivide the calculation scheme (4.51) into smaller, mutually exclusive subschemes. Using a decimated grid for the variable coefficient difference scheme amounts to rewriting scheme (4.51) as

$\displaystyle \begin{eqnarray}\hspace{-0.6in}I_{x,i+\frac{1}{2},j}(n+{\textstyl...
...{y,i,j-\frac{1}{2}}(n-{\textstyle \frac{1}{2}})\right) =\,\,\, 0 \end{eqnarray}$    

where we now compute solutions for $ i$, $ j$ and $ n$ integer. The interleaved grid is shown in Figure 4.18; a grey (white) dot at a grid location indicates that the adjacent named variable is to be calculated at times which are even (odd) multiples of $ T/2$. This interleaved form was originally put forth by Yee [214] in the context of electromagnetic field problems, and forms the basis of the widely used finite-difference time domain (FDTD) family of difference methods [184], which were discussed briefly in §4.1. If system (4.52) is rewritten as a TE or TM system, the interleaved arrangement of the field components also has an interesting physical interpretation as a discrete counterpart to the integral form of Ampere's and Faraday's Laws [184]. This result also extends easily to the discretization of Maxwell's equations in (3+1)D [214]; see §4.10.6 for more details.

Figure 4.18: Interleaved computational grid for the (2+1)D parallel-plate system.
% graphpaper(0,0)(250,25...
\put(230,185){\tiny {$I_{y}$}}
\end{picture} \par\end{center}\end{figure}

The (2+1)D analogue of (4.23), which holds in the case where $ l$ and $ c$ are constant, is

$\displaystyle \frac{v_{0}^{2}}{\gamma^{2}}\Big(U_{i,j}(n\!+\!1)+U_{i,j}(n\!-\!1)\Big) = U_{i+1,j}(n)+U_{i-1,j}(n)$ $\displaystyle +$ $\displaystyle U_{i,j+1}(n)+U_{i,j-1}(n)\notag$ (4.64)
  $\displaystyle +$ $\displaystyle 2\left(\frac{v_{0}^{2}}{\gamma^{2}}-2\right)U_{i,j}(n)$ (4.65)

The magic time step will now be

$\displaystyle v_{0} = \sqrt{2}\gamma$    

and (4.53) simplifies to

$\displaystyle U_{i,j}(n\!+\!1) + U_{i,j}(n\!-\!1) = \frac{1}{2}\Big(U_{i+1,j}(n)+U_{i-1,j}(n)+ U_{i,j+1}(n)+U_{i,j-1}(n)\Big)$ (4.66)

As in (1+1)D, when we are solving the wave equation by centered differences at the magic time step (or at CFL), the calculation further decomposes into two independent calculations; we need only update $ U_{i,j}(n)$ for $ i+j+n$ even (or odd). We will examine this interesting decomposition property in detail in Appendix A.

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Next: The Waveguide Mesh Up: The (2+1)D Parallel-plate System Previous: The (2+1)D Parallel-plate System
Stefan Bilbao 2002-01-22