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Next: Pseudopower and Pseudopassivity Up: Wave Digital Elements and Previous: The Bilinear Transform

Wave Variables

At this point, one may assume that we have finished; indeed, we can derive a discrete-time equivalent to any LTI $ N$-port (graphically represented by a signal flow diagram involving shifts and arithmetic operations), and such elements can be connected using Kirchoff's Laws, which remain unchanged by the mapping (2.11). In particular, a network consisting of a collection of connected passive $ N$-ports will possess a discrete equivalent of the passivity property, which has been called pseudopassivity [42]. The problem, however, is that a simple application of the bilinear transform to a given $ N$-port usually leaves us with port variables which are not related to each other in a strictly causal way. For example, the difference equation (2.13) that results in the case of the inductor relates $ v(n)$ to $ i(n)$ at every time step $ n$ so that if we try to connect such a discrete-time one-port to another which has the same property (using Kirchoff's Laws, which are memoryless), we necessarily end up with non-realizable delay-free loops [46] in our resulting signal flow diagram. In other words, we will not be able to explicitly update all the port variables in our algorithm using only past values stored in the delay registers.

The problem of these delay-free loops was solved by Fettweis [41] with the introduction of wave variables, a concept with a long history borrowed from microwave electronics [11,12]. For a port with voltage $ v$ and current $ i$, voltage waves are defined by

\begin{subequations}\begin{align}a &= v+iR &&\mbox{{\rm Input voltage wave}}\\ b &= v-iR &&\mbox{{\rm Output voltage wave}} \end{align}\end{subequations}

$ a$ and $ b$ are referred to as wave variables, and in particular, $ a$ is called an input wave and $ b$ an output wave; the significance of these names will become clear in the examples of §2.3.4. This definition holds instantaneously, and will also be true for continuous $ v$ and $ i$, though we will almost never have occasion to refer to analog wave variables in this thesis. The parameter $ R>0$ is a free parameter known as the port resistance--its choice is governed by the character of the element itself. We also can define the port conductance $ G$ by

$\displaystyle G = \frac{1}{R}$ (2.15)

at a port with port resistance $ R$.

It is also possible to define power-normalized waves [46] $ \underline{a}$ and $ \underline{b}$ at any port with port resistance $ R$ by

\begin{subequations}\begin{align}\underline{a} &= \frac{v+iR}{2\sqrt{R}}&&\mbox{...
...c{v-iR}{2\sqrt{R}}&&\mbox{{\rm Output power wave}} \end{align}\end{subequations}

The two types of waves are simply related to each other by
$\displaystyle \begin{eqnarray}a &=& 2\sqrt{R}\underline{a}\\ b &=& 2\sqrt{R}\underline{b} \end{eqnarray}$ (2.17a)

but power-normalized quantities have certain advantages in cases for which a port resistance is time-varying or signal dependent (indeed, in these cases, power-normalized waves must be employed if passivity in the digital simulation is to be maintained). In general, however, in view of (2.17), it should be assumed that we are using voltage waves unless otherwise indicated.

The steady state quantities $ \hat{a}$ and $ \hat{b}$ are defined in a manner identical to (2.14), where we replace $ v$ and $ i$ by $ \hat{v}$ and $ \hat{i}$.

next up previous
Next: Pseudopower and Pseudopassivity Up: Wave Digital Elements and Previous: The Bilinear Transform
Stefan Bilbao 2002-01-22