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Initial Conditions

We have dealt, so far, with a method of integrating the transmission line and parallel-plate transmission line systems, but have not examined the necessary initialization of the algorithm. We will deal, here, with the lossless source-free cases.

In (1+1)D, the hyperbolic system (4.17) requires two initial conditions. That is, we require the knowledge of initial current and voltage distributions along the line. We would like to enter the discrete equivalent of this data into the delay registers somehow at the first time step $ n=0$. From Figure 4.14, it should be clear that four sets of data are required: $ U_{x^{-},i}^{+}(0)$, $ U_{x^{+},i}^{+}(0)$, $ U_{c,i}^{+}(0)$, which are the initial incoming waves at the parallel junctions, and $ I_{c,i+\frac{1}{2}}^{+}({\textstyle \frac{1}{2}})$, the values initially stored in the self-loops at the series junctions.

The first problem we encounter is that, on our decimated grid, we calculate values of $ U_{J}$ and $ I_{J}$, the grid functions corresponding to voltage and current, at alternating time steps. We have chosen our grid such that for $ k$ and $ m$ half-integer, $ I_{k}(m)$ is calculated for odd values of $ 2m$, and $ U_{k}(m)$ for $ 2m$ even; at time zero, then, $ U_{J}$ is accessible (as a combination of wave variables). How then do we enter the current initial data into the algorithm? It turns out that this problem is rather simply addressed. We can do one of two things: set the value of $ I_{J}$ at time step $ \frac{1}{2}$ to be equal to a sampled version of $ i(x,0)$, and accept the error that this introduces, which will be $ O(T)$, or we can use any available numerical method (i.e. one which does not operate on a staggered grid) to propagate the initial data $ i_{0}(x)$ forward by $ T/2$. Such a method should be at least $ O(T^{2})$ accurate, but it is allowed to even be unstable, since we will only be using it to update once [176].

Assume then, that our initial data are $ U^{*}_{i}(0) = u(i\Delta,0)$, and $ I_{i+\frac{1}{2}}^{*}(\frac{1}{2})$, some approximation to $ i((i+{\textstyle \frac{1}{2}})\Delta, \frac{T}{2})$ obtained by either of the methods mentioned above. At time step $ n=0$, we can write the junction voltages $ U_{J,i}(0)$ as

$\displaystyle U_{J,i}(0) = \frac{2}{Y_{J,i}}\left(Y_{x^{+},i}U_{x^{+},i}^{+}(0)+Y_{x^{-},i}U_{x^{-},i}^{+}(0)+Y_{c,i}U_{c,i}^{+}(0)\right)$ (4.85)

and $ I_{J,i+\frac{1}{2}}(\frac{1}{2})$ as

$\displaystyle I_{J,i+\frac{1}{2}}({\textstyle \frac{1}{2}})$ $\displaystyle =$   $\displaystyle \frac{2}{Z_{J,i+\frac{1}{2}}}\left(Z_{x^{+},i+\frac{1}{2}}I_{x^{+...
...}})+Z_{c,i+\frac{1}{2}}I_{c,i+\frac{1}{2}}^{+}({\textstyle \frac{1}{2}})\right)$    
  $\displaystyle =$   $\displaystyle \frac{2}{Z_{J,i+\frac{1}{2}}}\left(U_{x^{+},i+\frac{1}{2}}^{+}({\...
...}})+Z_{c,i+\frac{1}{2}}I_{c,i+\frac{1}{2}}^{+}({\textstyle \frac{1}{2}})\right)$    
  $\displaystyle =$   $\displaystyle \frac{2}{Z_{J,i+\frac{1}{2}}}\left(-U_{x^{-},i+1}^{-}(0)+U_{x^{+}...
...(0)+Z_{c,i+\frac{1}{2}}I_{c,i+\frac{1}{2}}^{+}({\textstyle \frac{1}{2}})\right)$    
  $\displaystyle =$   $\displaystyle \frac{2}{Z_{J,i+\frac{1}{2}}}\left(-U_{J,i+1}(0)+U_{J,i}(0)+ U_{x...
...(0)+Z_{c,i+\frac{1}{2}}I_{c,i+\frac{1}{2}}^{+}({\textstyle \frac{1}{2}})\right)$ (4.86)

The safest and most general way of proceeding, given that the immittances $ Z_{c,i+\frac{1}{2}}$ and $ Y_{c,i}$ may be zero, depending on the type of network we are using, is to set the initial values in the self-loops to zero. In this case, we can set
\begin{subequations}\begin{alignat}{2} &U_{c,i}^{+}(0) &&= I_{c,i+\frac{1}{2}}^{...
...frac{1}{2}}^{*}({\textstyle \frac{1}{2}})\right) \end{alignat}\end{subequations}

It can be easily verified that with these initial values for the wave variables, the junction voltages $ U_{J,i}(0)$ and currents $ I_{J,i+\frac{1}{2}}({\textstyle \frac{1}{2}})$ calculated from the DWN by (4.73) and (4.74) respectively will be consistent with the initial values of the continuous problem to first order in $ \Delta$. These settings may be used with any of the three types of network mentioned in §4.3.6.

We may ask, however, whether there is a way of setting the initial values such that we achieve better initial accuracy. For a network of type I, say, we have $ Y_{c,i}=0$. Then, if $ Z_{c,i+\frac{1}{2}}$ is non-zero everywhere (this can always be arranged by operating slightly away from CFL), we may use

  $\displaystyle U_{x^{-},i}^{+}(0)$   $\displaystyle = U_{x^{+},i}^{+}(0) = \frac{1}{2}U_{i}^{*}(0)$    
  $\displaystyle I_{c,i+\frac{1}{2}}^{+}({\textstyle \frac{1}{2}})$   $\displaystyle = \frac{1}{2Z_{c,i+\frac{1}{2}}}\left(Z_{J,i+\frac{1}{2}}I_{i+\frac{1}{2}}^{*}({\textstyle \frac{1}{2}})-U_{i}^{*}(0)+U_{i+1}^{*}(0)\right)$    

in which case the waveguide network reproduces the initial currents and voltages with no error. Similarly, for a type II network, we may set

  $\displaystyle U_{x^{+},i}^{+}(0)$   $\displaystyle = U_{i}^{*}(0)-\frac{1}{2Y_{x^{+},i}}I_{i+\frac{1}{2}}^{*}({\textstyle \frac{1}{2}})$    
  $\displaystyle U_{x^{-},i}^{+}(0)$   $\displaystyle = U_{i}^{*}(0)+\frac{1}{2Y_{x^{-},i}}I_{i-\frac{1}{2}}^{*}({\textstyle \frac{1}{2}})$    
  $\displaystyle U_{c,i}^{+}(0)$   $\displaystyle = \frac{1}{2Y_{c,i}}\left((Y_{c,i}-Y_{x^{-},i}-Y_{x^{+},i})U_{i}^...
...\textstyle \frac{1}{2}})-I_{i-\frac{1}{2}}^{*}({\textstyle \frac{1}{2}})\right)$    

which also yields an exact calculation. Either of these two means of initializing wave variables may also be used in the type III DWN.

These initialization procedures generalize simply to (2+1)D, where the parallel-plate equations require three initial conditions $ u(x,y,0)$, $ i_{x}(x,y,0)$ and $ i_{y}(x,y,0)$. In general, we now have seven wave variables to set: the waves approaching any parallel junction with coordinates $ (i,j)$ at $ n=0$, namely $ U^{+}_{x^{-},i,j}(0)$, $ U^{+}_{x^{+},i,j}(0)$, $ U^{+}_{y^{-},i,j}(0)$, $ U^{+}_{y^{+},i,j}(0)$ and $ U^{+}_{c,i,j}(0)$, as well as the values stored in the self-loop registers at the series junctions, $ I^{+}_{x,c,i+\frac{1}{2},j}({\textstyle \frac{1}{2}})$ and $ I^{+}_{y,c,i,j+\frac{1}{2}}({\textstyle \frac{1}{2}})$. For the sake of brevity, we provide only the settings for the general case, analogous to (4.75):

  $\displaystyle U_{x^{-},i,j}^{+}(0)$   $\displaystyle = \frac{1}{4}\left(\frac{Y_{J,i,j}U_{i,j}^{*}(0)}{2Y_{x^{-},i,j}}+Z_{J,i-\frac{1}{2},j}I_{x,i-\frac{1}{2},j}^{*}({\textstyle \frac{1}{2}})\right)$    
  $\displaystyle U_{x^{+},i,j}^{+}(0)$   $\displaystyle = \frac{1}{4}\left(\frac{Y_{J,i,j}U_{i,j}^{*}(0)}{2Y_{x^{+},i,j}}-Z_{J,i+\frac{1}{2},j}I_{x,i+\frac{1}{2},j}^{*}({\textstyle \frac{1}{2}})\right)$    
  $\displaystyle U_{y^{-},i,j}^{+}(0)$   $\displaystyle = \frac{1}{4}\left(\frac{Y_{J,i,j}U_{i,j}^{*}(0)}{2Y_{y^{-},i,j}}+Z_{J,i,j-\frac{1}{2}}I_{y,i,j-\frac{1}{2}}^{*}({\textstyle \frac{1}{2}})\right)$    
  $\displaystyle U_{y^{+},i,j}^{+}(0)$   $\displaystyle = \frac{1}{4}\left(\frac{Y_{J,i,j}U_{i,j}^{*}(0)}{2Y_{y^{+},i,j}}-Z_{J,i,j+\frac{1}{2}}I_{y,i,j+\frac{1}{2}}^{*}({\textstyle \frac{1}{2}})\right)$    

with, in addition,

$\displaystyle U_{c,i,j}^{+}(0) = I_{x,i+\frac{1}{2},j}^{+}({\textstyle \frac{1}{2}}) = I_{y,i,j+\frac{1}{2}}^{+}({\textstyle \frac{1}{2}}) = 0$    


  $\displaystyle U_{i,j}^{*}(0)$   $\displaystyle = u(i\Delta,j\Delta,0)$    
  $\displaystyle I_{x,i+\frac{1}{2},j}^{*}({\textstyle \frac{1}{2}})$   $\displaystyle = i_{x}((i+{\textstyle \frac{1}{2}})\Delta,j\Delta,{\textstyle \frac{1}{2}})$    
  $\displaystyle I_{y,i,j+\frac{1}{2}}^{*}({\textstyle \frac{1}{2}})$   $\displaystyle = i_{y}(i\Delta,(j+{\textstyle \frac{1}{2}})\Delta,{\textstyle \frac{1}{2}})$    

Because DWNs of the forms discussed in the previous sections are equivalent to two-step finite difference methods, problems with parasitic modes do not arise as they do in wave digital networks which simulate the same systems. This problem was discussed in detail in §3.9.

next up previous
Next: Alternative Grids in (2+1)D Up: Digital Waveguide Networks Previous: Grid Arrangement with Normal
Stefan Bilbao 2002-01-22