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Consider the waveguide network pictured in Figure 4.8. Each scattering junction (in this case parallel) is connected to its two neighbors by unit sample bidirectional delay lines. The spacing of the junctions is
and the waveguide delays are of duration
. The voltage at a junction with coordinate
and at time
is denoted by
for integer
and ![$ n$](img74.png)
.
Figure 4.8:
(1+1)D waveguide network.
![\begin{figure}\begin{center}
\begin{picture}(545,150)
% graphpaper(0,0)(545,150...
...all {Y_{x^{+}\!,i} = Y_{x^{-}\!,i+1}}}$}
\end{picture} \end{center} \end{figure}](img1262.png) |
We can name the voltages and current flows in individual waveguides in the following way. At junction
, the line voltages are:
|
![$\displaystyle U_{x^{+},i}$](img1263.png) |
|
voltage in waveguide leading east![$\displaystyle \notag$](img1264.png) |
|
|
![$\displaystyle U_{x^{-},i}$](img1265.png) |
|
voltage in waveguide leading west![$\displaystyle \notag$](img1264.png) |
|
and the flows are:
|
![$\displaystyle I_{x^{+},i}$](img1266.png) |
|
current flow in waveguide leading east![$\displaystyle \notag$](img1264.png) |
|
|
![$\displaystyle I_{x^{-},i}$](img1267.png) |
|
current flow in waveguide leading west![$\displaystyle \notag$](img1264.png) |
|
The constraints, imposed by Kirchoff's Laws at a parallel junction, are:
![$\displaystyle U_{J,i} = U_{x^{+},i} = U_{x^{-},i}\hspace{1.0in} I_{x^{+},i} + I_{x^{-},i} = 0$](img1268.png) |
(4.25) |
As discussed in §4.2, the voltages and current flows in the individual waveguides can be further broken up into incoming and outgoing waves. That is, we have, at a junction at grid location
:
where
is either of
or
. The variables superscripted with a
refer to the incoming waves, and those marked
to outgoing waves. In a particular waveguide section, the current and voltage waves are related by:
![$\displaystyle I_{q,i}^{+} = Y_{q,i}U_{q,i}^{+}\hspace{1.0in}I_{q,i}^{-} = -Y_{q,i}U_{q,i}^{-}$](img1272.png) |
(4.26) |
where
is the characteristic admittance of the waveguide connected to junction
in direction
. In addition, because the junctions at
and
are connected to opposite ends of the same waveguide, we have
As before, we will also define the impedance of any waveguide to be
At a particular parallel junction, the junction admittance will thus be
In this case, from (4.14), the junction voltage can be written in terms of incoming wave variables as
![$\displaystyle U_{J,i} = \frac{2}{Y_{J,i}}\left(Y_{x^{-},i}U_{x^{-},i}^{+}+Y_{x^{+},i}U_{x^{+},i}^{+}\right)$](img1278.png) |
(4.27) |
and the outgoing voltage waves from any junction are related to the incoming waves by
where
refers to either of the directions
or
.
The incoming voltage wave entering each junction from a particular waveguide at time step
is simply the outgoing voltage wave leaving a neighboring junction, one time step before. Reading directly from Figure 4.8, we have
![$\displaystyle \begin{eqnarray}U_{x^{+},i}^{+}(n) &=& U_{x^{-},i+1}^{-}(n-1)\\ U_{x^{-},i}^{+}(n) &=& U_{x^{+},i-1}^{-}(n-1) \end{eqnarray}$](img1280.png) |
(4.28a) |
The case of flow waves is similar except for a sign inversion--that is, we have
![$\displaystyle \begin{eqnarray}I_{x^{+},i}^{+}(n) &=& -I_{x^{-},i+1}^{-}(n-1)\\ I_{x^{-},i}^{+}(n) &=& -I_{x^{+},i-1}^{-}(n-1) \end{eqnarray}$](img1281.png) |
(4.29a) |
As discussed in §4.2, we can perform all calculations using voltage waves; in the waveguide networks pictured in this chapter, we will always assume, without loss of generality, that we are dealing with voltage waves.
Next: Waveguide Network and the
Up: The (1+1)D Transmission Line
Previous: Centered Difference Schemes and
Stefan Bilbao
2002-01-22