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A (1+1)D Waveguide Network

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 $ \Delta$ and the waveguide delays are of duration $ T$. The voltage at a junction with coordinate $ i\Delta$ and at time $ nT$ is denoted by $ U_{J,i}(n)$ for integer $ i$ and $ n$% latex2html id marker 82641

Figure 4.8: (1+1)D waveguide network.
% graphpaper(0,0)(545,150...
...all {Y_{x^{+}\!,i} = Y_{x^{-}\!,i+1}}}$}
\end{picture} \end{center} \end{figure}

We can name the voltages and current flows in individual waveguides in the following way. At junction $ i$, the line voltages are:

  $\displaystyle U_{x^{+},i}$   $\displaystyle =$   voltage in waveguide leading east$\displaystyle \notag$    
  $\displaystyle U_{x^{-},i}$   $\displaystyle =$   voltage in waveguide leading west$\displaystyle \notag$    

and the flows are:

  $\displaystyle I_{x^{+},i}$   $\displaystyle =$   current flow in waveguide leading east$\displaystyle \notag$    
  $\displaystyle I_{x^{-},i}$   $\displaystyle =$   current flow in waveguide leading west$\displaystyle \notag$    

The constraints, imposed by Kirchoff's Laws at a parallel junction, are:

$\displaystyle U_{J,i} = U_{x^{+},i} = U_{x^{-},i}\hspace{1.0in} I_{x^{+},i} + I_{x^{-},i} = 0$ (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 $ i$:

$\displaystyle U_{q,i} = U_{q,i}^{+}+U_{q,i}^{-}\hspace{1.0in}I_{q,i} = I_{q,i}^{+}+I_{q,i}^{-}$    

where $ q$ is either of $ x^{+}$ or $ x^{-}$. 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:

$\displaystyle I_{q,i}^{+} = Y_{q,i}U_{q,i}^{+}\hspace{1.0in}I_{q,i}^{-} = -Y_{q,i}U_{q,i}^{-}$ (4.26)

where $ Y_{q,i}$ is the characteristic admittance of the waveguide connected to junction $ i$ in direction $ q$. In addition, because the junctions at $ i$ and $ i+1$ are connected to opposite ends of the same waveguide, we have

$\displaystyle Y_{x^{-},i+1} = Y_{x^{+},i}$    

As before, we will also define the impedance of any waveguide to be

$\displaystyle Z_{q,i} = \frac{1}{Y_{q,i}}$    

At a particular parallel junction, the junction admittance will thus be

$\displaystyle Y_{J,i} \triangleq Y_{x^{+},i}+Y_{x^{-},i}$    

In this case, from (4.14), the junction voltage can be written in terms of incoming wave variables as

$\displaystyle U_{J,i} = \frac{2}{Y_{J,i}}\left(Y_{x^{-},i}U_{x^{-},i}^{+}+Y_{x^{+},i}U_{x^{+},i}^{+}\right)$ (4.27)

and the outgoing voltage waves from any junction are related to the incoming waves by

$\displaystyle U_{r,i}^{-} = -U_{r,i}^{+}+U_{J,i}$    

where $ r$ refers to either of the directions $ x^{+}$ or $ x^{-}$.

The incoming voltage wave entering each junction from a particular waveguide at time step $ n$ is simply the outgoing voltage wave leaving a neighboring junction, one time step before. Reading directly from Figure 4.8, we have

$\displaystyle \begin{eqnarray}U_{x^{+},i}^{+}(n) &=& U_{x^{-},i+1}^{-}(n-1)\\ U_{x^{-},i}^{+}(n) &=& U_{x^{+},i-1}^{-}(n-1) \end{eqnarray}$ (4.28a)

The case of flow waves is similar except for a sign inversion--that is, we have
$\displaystyle \begin{eqnarray}I_{x^{+},i}^{+}(n) &=& -I_{x^{-},i+1}^{-}(n-1)\\ I_{x^{-},i}^{+}(n) &=& -I_{x^{+},i-1}^{-}(n-1) \end{eqnarray}$ (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 up previous
Next: Waveguide Network and the Up: The (1+1)D Transmission Line Previous: Centered Difference Schemes and
Stefan Bilbao 2002-01-22