Multi-Dimensional Elements and Wave Digital Filters

MD N-ports

Need to extend the notion of passivity to distributed networks.

Basic element, the $N$ -port has the same form in multi-D:

• ``Port'' is no longer localized in space, and thus we have, for any port $j$ ,
  

  $$v_{j} = v_{j}({\bf x},t)$$

  $$i_{j} = i_{j}({\bf x},t)$$

  
  

• instantaneous power (or power density) absorbed at any point $({\bf x},t)$ is

  $$p_{inst}({\bf x},t) = \sum_{j=1}^{N}v_{j}i_{j}$$

  
  

• Also have distributed source power $p_{s}$ and dissipated power $p_{d}$
• $N$ -ports must be connected portwise
• stored energy requires a generalized treatment...

MD N-ports Continued

• If the $N$ -port is reactive, we have an associated stored energy flux. In terms of causal coordinates ${\bf t} = (t_{1}\hdots t_{k})$ , this is a vector function

  $$W_{st}({\bf x},t) = (W_{1}, W_{2}, \hdots, W_{k})$$

  
  
  For an causal $MD-passive$ $N$ -port defined over a region $G$ , we need $W_{s}\geq 0$ everywhere in $G$ . Stored energy flows forward in physical time (and thus forward in all the causal coordinates)
  
  
  
• Integral energy balance (with respect to a region $G$ ) is

  $$\int_{G}\left(p_{inst}+p_{s}\right)d{\bf t} = \int_{G}p_{d}d{\bf t} + \int_{\partial G} W_{st}\cdot{\bf n}d\sigma$$

  
  

MD-Passivity

An MD $N$ -port is called integrally $MD$ -passive (in $G$ ) if we have

$$\int_{G}p_{inst}d{\bf t} - \int_{\partial G} W_{st}\cdot{\bf n}d\sigma\geq 0$$

or differentially $MD$ passive, if we have

$$p_{inst} - \nabla\cdot W_{st}\geq 0$$

and $MD$ -lossless if equality holds.

• Kirchoff's Laws (treated as an $N$ -port) are the same in $MD$ , so that a Kirchoff connection of $MD$ $N$ -ports will itself behave passively (Tellegen).
• The resistor, defined by $v=iR$ , $R$ constant is also the same in $MD$ , so $MD$ -passive as well.
• Transformers, gyrators lossless as well, provided we define $W_{st}\equiv {\bf0}$
• Reactive elements need a more involved treatment...

The MD Inductor

Consider the following relation:

$$v({\bf t}) = L\frac{\partial i({\bf t})}{\partial t_{j}}$$

where $L$ is a positive constant, and $t_{j}, j=1\hdots k$ is an causal coordinate.

• Defined over entire ${\bf t}$ space, but is really just a set of $1D$ differential relations.
• $v$ obtained from $i$ by integrating in $t_{j}$ direction (forward in time).

We can define a discrete approximation by applying the trapezoid rule in the $t_{j}$ direction:

$$\frac{v_{\hdots,n_{j},\hdots}+v_{\hdots, n_{j}-1,\hdots}}{2} = \frac{L}{T_{1}}\left(i_{\hdots,n_{j},\hdots}-i_{\hdots, n_{j}-1,\hdots}\right)$$

And introducing wave variables,

$$a_{n_{1},\hdots,n_{k}} = v_{n_{1},\hdots,n_{k}}+Ri_{n_{1},\hdots,n_{k}}$$

$$b_{n_{1},\hdots,n_{k}} = v_{n_{1},\hdots,n_{k}}-Ri_{n_{1},\hdots,n_{k}}$$

can get the MD-equivalent of the wave-digital inductance one-port:

$$b_{\hdots,n_{j},\hdots}=-a_{\hdots,n_{j}-1,\hdots}, \hspace{0.5in}, R = \frac{2L}{T_{j}}$$ (1)